# Modeling Multi-Granularity Hierarchical Features for Relation Extraction

Xinnian Liang1111Contribution during internship at Tencent Inc., Shuangzhi
Wu2, Mu Li2 Zhoujun Li1222Corresponding Author
1State Key Lab of Software Development Environment, Beihang University,
Beijing, China
2Tencent Cloud Xiaowei, Beijing, China
<EMAIL_ADDRESS>frostwu<EMAIL_ADDRESS>

###### Abstract

Relation extraction is a key task in Natural Language Processing (NLP), which
aims to extract relations between entity pairs from given texts. Recently,
relation extraction (RE) has achieved remarkable progress with the development
of deep neural networks. Most existing research focuses on constructing
explicit structured features using external knowledge such as knowledge graph
and dependency tree. In this paper, we propose a novel method to extract
multi-granularity features based solely on the original input sentences. We
show that effective structured features can be attained even without external
knowledge. Three kinds of features based on the input sentences are fully
exploited, which are in entity mention level, segment level, and sentence
level. All the three are jointly and hierarchically modeled. We evaluate our
method on three public benchmarks: SemEval 2010 Task 8, Tacred, and Tacred
Revisited. To verify the effectiveness, we apply our method to different
encoders such as LSTM and BERT. Experimental results show that our method
significantly outperforms existing state-of-the-art models that even use
external knowledge. Extensive analyses demonstrate that the performance of our
model is contributed by the capture of multi-granularity features and the
model of their hierarchical structure. Code and data are available at
https://github.com/xnliang98/sms.

##  1 Introduction

Relation extraction (RE) is a fundamental task in Natural Language Processing
(NLP), which aims to extract relations between entity pairs from given plain
texts. RE is the cornerstone of many downstream NLP tasks, such as knowledge
base construction Ji and Grishman (2011), question answering Yu et al. (2017),
and information extraction Fader et al. (2011).

Most recent works focus on constructing explicit structured features using
external knowledge such as knowledge graph, entity features and dependency
tree. To infuse prior knowledge from existing knowledge graph, recent works
Peters et al. (2019a); Wang et al. (2020b, a) proposed some pre-train tasks to
help model learn and select proper prior knowledge in the pre-training stage.
Baldini Soares et al. (2019); Yamada et al. (2020); Peng et al. (2020) force
model learning entitiy-related information via well-designed pre-train tasks.
Zhang et al. (2018); Guo et al. (2019); Xue et al. (2020); Chen et al. (2020)
encode dependency tree with graph neural network Kipf and Welling (2017) (GNN)
to help RE models capture non-local syntactic relation. All of them achieve a
remarkable performance via employing external information from different
structured features.

However, they either need time-consuming pre-training with external knowledge
or need an external tool to get a dependency tree which may introduce
unnecessary noise. In this paper, we aim to attain effective structured
features based solely on the original input sentences. To this end, we analyze
previous typical works and find that three kinds of features mainly affect the
performance of RE models, which are entity mention level111entity mentions
contain the entity itself and co-references of it., segment222continuous words
in sentence (n-gram) level and sentence level features. Sentence level and
entity mention Baldini Soares et al. (2019); Yamada et al. (2020); Peng et al.
(2020) level features were widely used by previous works but segment level
feature Yu et al. (2019); Joshi et al. (2020) does not get as much attention
as the previous two features. These three level features can provide different
granularity information from input sentences for relation prediction Chowdhury
and Lavelli (2012); Kim . However, recent works did not consider them at the
same time and ignored the structure and interactive of them.

Figure 1: An Example for relation extraction which shows the hierarchical
structure between entity mention level and segment level features.

We employ a simple example in Figure 1 to show the hierarchical and joint
structure of the previous three granularities features. The hierarchical
structure is between mention level and segment level features. This example
gives sentence and entity pairs (“Steven Jobs", “Apple"). We can find that the
relation “the_CEO_of" of given entity “Apple" and entity “Steven Jobs" is
built upon the core segment “the CEO of" between the entity mention “it" (i.e.
co-reference of entity “Apple") and entity “Steven Jobs". To extract relation
from this example, RE models need to first capture mention level features of
given entities and then catch core segment level feature “the CEO of" which is
related to mention level features. Finally, RE models can easily predict the
relation with previous two-level hierarchical features. Besides, sentence-
level semantic features can assist RE models to predict the relation of
examples without an explicit pattern of entity mentions and segments.

Following previous intuitive process, we propose a novel method which extracts
multi-granularity features based solely on the original input sentences.
Specifically, we design a hierarchical mention-aware segment attention, which
employs a hierarchical attention mechanism to build association between entity
mention level and segment level features. Besides, we employ a global semantic
attention to get a deeper understanding of sentence level features from input
sentence representation. Finally, we aggregate previously extracted multi-
granularity features with a simple fully-connected layer to predict the
relation.

To evaluate the effectiveness of our method, we combine our method with
different text encoders (e.g. LSTM and BERT) and results show that our method
can bring significant improvement for all of them. Compared with models
without using external knowledge, SpanBERT with our method can achieve a new
state-of-the-art result on Semeval 2010 Task 8, Tacred and Tacred Revisited.
It is deserved to mention that the performance of our model is very
competitive with the state-of-the-art models, which employ large-scale extra
training data or information. We also do many analyses to demonstrate that
features from representation of input itself are enough for the sentence-level
RE tasks and multi-granularity features with hierarchical structure are
crucial for relation prediction.

##  2 Methodology

Figure 2: Middle: The structure of our proposed multi-granularity hierarchical
feature extractor. Left: Details of global semantic attention (sentence level
feature) and feature aggregation layer. Right: Details of mention attention
(entity mention level feature) and mention-aware segment attention (segment
level feature).

The structure of our model and details of each component is shown in figure 2.
We can see the overall architecture in the middle. It is divided into three
components from bottom to top: 1) A text encoder which is employed to obtain
text vector representations; 2) A multi-granularity hierarchical feature
extractor which can exploit effective structured features from text
representations; 3) A feature aggregation layer which aggregate previous
multi-granularity features for relation prediction. In this section, we will
introduce details of three components.

Firstly, we formalize the relation extraction task. Let
$\mathtt{x}=\\{x_{1},x_{2},...,x_{n}\\}$ be a sequence of input tokens, where
$x_{0}=[\mathtt{CLS}]$ and $x_{n}=[\mathtt{SEP}]$ are special start and end
tokens for BERT-related encoders. Let $s_{1}=(i,j)$ and $s_{2}=(k,l)$ be pairs
of entity indices. The indices in $s_{1}$ and $s_{2}$ delimit entities in
$\mathtt{x}$: $[x_{i},\dots,x_{j-1}]$ and $[x_{k},\dots,x_{l-1}]$. Our goal is
to learn a function $P(r)=f_{\theta}(\mathtt{x},s_{1},s_{2})$, where
$r\in\mathcal{R}$ indicates the relation between the entity pairs, which is
marked by $s1$ and $s2$. $\mathcal{R}$ is a pre-defined relation set.

###  2.1 Encoder Layer

We first employ a text encoder (e.g. BERT) to map tokens in input sentences
into vector representations which can be formalized by Equ. (1).

$\mathbf{H}=\\{h_{0},\dots,h_{n}\\}=f_{encoder}(x_{0},\dots,x_{n})$ (1)

Where $\mathbf{H}=\\{h_{0},\dots,h_{n}\\}$ is the vector representation of
input sentences.

Our work is built upon $\mathbf{H}$ and does not need any external
information. We employ a max-pooling operation to obtain shallow features of
entity pairs and input sentences. $h_{e_{1}}=\mathtt{Maxpooling}(h_{i:j})$ and
$h_{e_{2}}=\mathtt{Maxpooling}(h_{k:l})$ are the representations of entity
pairs. $h_{g}=\mathtt{Maxpooling}(\mathbf{H})$ is the vector representation of
input sentences which contains global semantic information.

###  2.2 Multi-Granularity Hierarchical Feature Extractor

The multi-granularity hierarchical feature extractor is the core component of
our method and it consists of three attention mechanism for different
granularity features extraction: 1) mention attention which is designed to
entity mention features of given entity pairs; 2) mention-aware segment
attention which is based on the entity mention features from previous mention
attention and aim to extract core segment level feature which is related to
entity mentions; 3) global semantic attention which focuses on the sentence
level feature.

####  2.2.1 Mention Attention

The structure of mention attention is shown in the right bottom of Figure 2.
To capture more information about given entity pairs from input sentences, we
extract entity mention level features by modeling the co-references (mentions)
of entities implicitly. We employ a mention attention to capture information
about entity 1 and 2 respectively. Specifically, we can use the representation
of an entity as a query to obtain the entity mention feature from $\mathbf{H}$
by Equ. (2).

$\displaystyle h^{\prime}_{e_{1}}$
$\displaystyle=\mathtt{Softmax}(\frac{\mathbf{H}\cdot
h_{e_{1}}}{\sqrt{d}})\cdot\mathbf{H}$ (2) $\displaystyle h^{\prime}_{e_{2}}$
$\displaystyle=\mathtt{Softmax}(\frac{\mathbf{H}\cdot
h_{e_{2}}}{\sqrt{d}})\cdot\mathbf{H}$

Where $d$ is the dimension of vector representation and used to normalize
vectors. Then, $h^{\prime}_{e_{1}}$ and $h^{\prime}_{e_{2}}$ model the
mentions of given entity pairs implicitly and contain more entity semantic
information than $h_{e_{1}}$ and $h_{e_{2}}$.

####  2.2.2 Mention-Aware Segment Attention

The structure of mention-aware segment attention is shown in the right top of
Figure 2. And the mention-aware segment attention is a hierarchical structure
based on the entity mention features $h^{\prime}_{e_{1}}$ and
$h^{\prime}_{e_{2}}$ from mention attention.

Before introducing mention-aware segments attention, we first introduce how to
get the representations of segments. We employ convolutional neural networks
(CNN) with different kernel sizes to obtain all n-gram segments in texts,
which can effectively capture local n-gram information with Equ. (3).

$\mathbf{H}_{t}=\mathtt{CNN}_{t}(\mathbf{H}),t\in\\{1,2,3\\}$ (3)

Where $t$ is the kernel size of CNN and is empirically set as
$t\in\\{1,2,3\\}$ which means extract 1-gram, 2-gram ,and 3-gram segment level
features.

Intuitively, the valuable segments should be highly related to given entity
pairs, which can help the model to decide the relation of given entity pairs.
Entity mention features $h^{\prime}_{e_{1}}$ and $h^{\prime}_{e_{2}}$ contain
comprehensive information of given entity pairs and $\mathbf{H}_{t}$ contain
1,2,3-gram segment level features. We can extract mention-aware segment level
features by simply linking them with attention mechanisms by Equ. (4).

$h_{m}^{t}=\mathtt{Softmax}(\frac{\mathbf{H}_{t}\cdot(W_{m}[h^{\prime}_{e_{1}};h^{\prime}_{e_{2}}])}{\sqrt{d}})\cdot\mathbf{H}_{t}$
(4)

Then, we get $\\{h_{m}^{t}\\}_{t=1,2,3}$ which capture different granularity
segments features.

####  2.2.3 Global Semantic Attention

The structure of global semantic attention is shown in the left bottom of
Figure 2. Previous works always directly concatenate vector representation
$[h_{e_{1}};h_{e_{2}};h_{g}]$ as the global semantic feature of input text. We
argue this is not enough to help model capture deeper sentence level semantic
information for RE. Different from them, to obtain better global sentence-
level semantic feature, we employ an attention operation called global
semantic attention which use the concatenation of
$[h_{e_{1}};h_{e_{2}};h_{g}]$ as query to capture deeper semantic feature from
context representation $\mathbf{H}$ by Equ. (5).

$h_{s}=\mathtt{Softmax}(\frac{\mathbf{H}\cdot(W_{s}[h_{e_{1}};h_{e_{2}};h_{g}])}{\sqrt{d}})\cdot\mathbf{H}$
(5)

Where $W_{s}\in\mathbb{R}^{d\times 3d}$ is a linear transform matrix, and $d$
is a hidden dimension of vectors. The concatenation of
$[h_{e_{1}};h_{e_{2}};h_{g}]$ is used as a query of the attention operation,
which can force the extracted global semantic representation $h_{s}$ contain
entity mention related sentence level feature.

###  2.3 Feature Aggregation Layer

The structure of the feature aggregation layer is shown in the left top of
Figure 2. We aggregate previous multi-granularity features by Equ. (6).

$h_{o}=\mathtt{ReLU}(W_{a}[h_{s};h^{\prime}_{e_{1}};h^{\prime}_{e_{1}};h_{m}^{1};;h_{m}^{2};;h_{m}^{3}])$
(6)

Where $W_{a}\in\mathbb{R}^{6d\times d}$ is a linear transform matrix and
$\mathtt{ReLU}$ is a nonlinear activation function.

###  2.4 Classification

Finally, we employ a softmax function to output the probability of each
relation label as follows:

$P(r|\mathtt{x},s_{1},s_{2})=\mathtt{Softmax}(W_{o}h_{o})$ (7)

The whole model is trained with cross entropy loss function. We call the
multi-granularity hierarchical feature extractor: SMS (relation extraction
with Sentence level, Mention level and mention-aware Segment level features).

##  3 Experiments

| Tacred | Semeval
---|---|---
lr | 3e-5 | 2e-5
warmup steps | 300 | 0
batch size | 64 | 32
V100 GPU | 4x | 1x
epochs | 4 | 10
max length | 128 | 128
Table 1: Hyper-parameters used in training. | Model | Tacred | Tacred Revisited
---|---|---|---
| P($\Delta P$) | R($\Delta R$) | F1($\Delta F1$) | P($\Delta P$) | R($\Delta R$) | F1($\Delta F1$)
1 | LSTM | 62.5 | 63.4 | 62.9 | 67.7 | 73.1 | 70.3
PA-LSTM* | 65.7 | 64.5 | 65.1 | 73.6 | 72.8 | 73.2
SA-LSTM | 68.1 | 65.7 | 66.9 | 78.3 | 72.5 | 75.4
C-GCN* | 69.9 | 63.3 | 66.4 | 76.8 | 71.4 | 74.1
C-AGGCN* | 71.9 | 63.4 | 67.5 | 78.2 | 70.5 | 74.3
2 | TRE | - | - | 67.4 | - | - | 75.3
BERT-base | 68.1 | 67.7 | 67.9 | 69.4 | 75.8 | 72.6
BERT-large | 69.2 | 69.4 | 69.3 | 75.1 | 74.8 | 75.0
BERT+LSTM | 73.3 | 63.1 | 67.8 | 74.1 | 73.9 | 74.0
SpanBERT-base | 67.6 | 68.6 | 68.1 | 69.1 | 78.2 | 73.7
SpanBERT-large | 70.8 | 70.9 | 70.8 | 77.8 | 78.3 | 78.0
DG-SpanBERT-large* | 71.4 | 71.6 | 71.5 | 79.2 | 78.6 | 78.9
3 | MTB† | - | - | 71.5 | - | - | -
KnowBERT-W+W‡ | 71.6 | 71.4 | 71.5 | 79.0 | 79.6 | 79.3
KEPLER‡ | 70.4 | 73.0 | 71.7 | - | - | -
K-Adapter‡ | 70.1 | 74.0 | 72.0 | - | - | -
LUKE† | 70.4 | 75.1 | 72.7 | 79.7 | 80.6 | 80.2
4 | LSTM+SMS | 72.8(+10.3) | 64.6(1.2) | 68.4(+5.5) | 80.8(+3.1) | 72.2(-0.9) | 75.9(+5.6)
SpanBERT-base+SMS | 72.6(+5.0) | 68.4(-0.2) | 70.5(+2.4) | 79.1(+10.0) | 77.7(-0.5) | 78.3(+4.6)
SpanBERT-large+SMS | 72.2(+1.4) | 71.6(+0.7) | 71.9(+1.1) | 79.3(+1.5) | 80.4(+2.1) | 79.8(+1.8)
BERT-base+SMS | 69.4(+1.3) | 70.2(+2.5) | 69.7(+1.8) | 77.0(+7.6) | 80.1(+4.3) | 78.5(+5.9)
BERT-large+SMS | 70.7(+1.5) | 69.1(-0.3) | 69.9(+0.6) | 78.9(+3.8) | 79.2(+4.4) | 79.1(+4.1)
Table 2: Results on Tacred and Tacred Revisited. Bold means the best results
in each block. Underline means the best results in block 1, 2, and 4. * means
that the model employs dependency tree information. †means that the model
employs extra training data to pre-train the model. ‡means the model employs
knowledge graphs.

###  3.1 Datasets

We evaluate the performance of our method on Semeval 2010 Task 8, Tacred and
Tacred Revisited datasets.

SemEval 2010 Task 8 Hendrickx et al. (2010) is a public dataset which contains
10,717 instances with 9 relations. The training/validation/test set contains
7,000/1,000/2,717 instances respectively.

Tacred333https://nlp.stanford.edu/projects/tacred/ is one of the largest, most
widely used crowd-sourced datasets for Relation Extraction (RE), which is
introduced by Zhang et al. (2017), with 106,264 examples built over newswire
and web text from the corpus used in the yearly TAC Knowledge Base Population
(TAC KBP) challenges. The training/validation/test set contains
68,124/22,631/15,509 instances respectively. It covers 42 relation types
including 41 relation types and a no_relation type and contains longer
sentences with an average sentence length of 36.4.

Tacred Revisited444https://github.com/DFKI-NLP/tacrev was proposed by Alt et
al. (2020) which aims to improve the accuracy and reliability of future RE
method evaluations. They validate the most challenging 5K examples in the
development and test sets using trained annotators and find that label errors
account for 8% absolute F1 test error, and that more than 50% of the examples
need to be relabeled. Then, they relabeled the test set and released the
Tacred Revisited dataset.

###  3.2 Settings

The setting of hyper-parameters is shown in table 1. Following the
implementation details mentioned in Zhang et al. (2017), we employ the “entity
mask” strategy and the “multi-channel” strategy during experiments. The former
means replacing each subject entity (and object entity similarly) in the
original sentence with a special [SUBJ-$\langle$NER$\rangle$] token. All of
our reported results are the mean of 5 results with different seeds, which are
randomly selected. We evaluate the models on Tacred with the official
script555https://github.com/yuhaozhang/tacred-relation in terms of the
Macro-F1 score and on Semeval with the official script
semeval2010_task8_scorer-v1.2.pl.

When employing LSTM as the encoder, we employ a single-layer bidirectional
LSTM with the hidden dimension size set to 200, we set dropout after the input
layer and before the output layer with $p$ = 0.5. We use stochastic gradient
descent (SGD) with epochs of 30, learning rate of 1.0, decay weight of 0.5 and
batch sizes of 50 to train the model. The latter is to augment the input by
concatenating it with part-of-speech (POS) and named entity recognition (NER)
embeddings. Glove Pennington et al. (2014) embedding with 300-dimension is
used for initializing word embedding layers in LSTM+SMS. NER embedding, POS
embedding and position embedding are randomly initialized with 30-dimension
vectors from uniform distribution.

###  3.3 Comparison Models

We mainly compare with models which are based on pre-trained language models
(e.g. BERT). We reproduce the results of BERT and SpanBERT to evaluate the
improvement of our method. We also compared other models with pre-trained
language models. TRE Alt et al. (2019), which uses the unidirectional OpenAI
Generative Pre-Trained Transformer (GPT) Radford et al. (2019). BERT-LSTM Shi
and Lin (2019), which stacks bidirectional LSTM layer to BERT encoder. DG-
SpanBERT, which replaced the encoder of C-GCN Zhang et al. (2018) with
SpanBERT and achieved the new state-of-the-art result without extra training
data. MTB Baldini Soares et al. (2019), which incorporates an intermediate
“matching the blanks” pre-training on the entity-linked text based on English
Wikipedia. KnowBERT-W+W Peters et al. (2019b), which is an an advanced version
of KnowBERT. KEPLER Wang et al. (2020b), which integrates factual knowledge
with the supervision of the knowledge embedding objective. K-Adapter Wang et
al. (2020a), which consists of a RoBERTa model and an adapter to select
adaptive knowledge. LUKE Yamada et al. (2020), which is trained with a new
pre-training task which involves predicting randomly masked words and entities
in a large entity-annotated corpus retrieved from Wikipedia and introduce a
new entity-aware attention mechanism.

In order to further prove the effectiveness of our SMS, we use bi-directional
LSTM as encoder, and compare with models which do not use pre-trained language
models. We choose two sequence-based models. PA-LSTM Zhang et al. (2017),
which employ Bi-LSTM to encoder the plain text and combine with position-aware
attention mechanism to extract relation. PA-SLTM is the benchmark of Tacred.
SA-LSTM Yu et al. (2019), which employ CRF to learn segment-level attention
and is the best sequence-based model of Tacred.

We also compare our model with two other dependency-based models which make
use of GCN Kipf and Welling (2017) to capture semantic information from the
dependency tree. C-GCN Zhang et al. (2018), which applies pruning strategy and
GCN to extract features from tree structure for relation extraction. C-AGGCN
Guo et al. (2019), which introduces self-attention to build a soft adjacent
matrix as input of Dense GCN to learn tree structure features.

###  3.4 Results on Tacred and Tacred Revisited

We first report the results or our model on Tacred and Tacred Revisited on
Table 2. Compared models are divided into three categories: 1) models with Bi-
LSTM encoder in block 1; 2) models with pre-trained models in block 2; 3)
models with external knowledge in block 3. The results of our model are
reported in block 4. We use * to mark models with dependency trees which are
obtained with external tools. We use †to mark models which use external
training data to pre-train the model and ‡to mark models which employ
knowledge graphs to pre-train or fine-tune the model. Models with †and
‡require external data and we do not directly compare them.

####  3.4.1 Compare with Pre-trained models

We can see that our SMS can bring at least 0.6 and up to 5.5 F1 score
improvement for the original encoder on Tacred dataset. On the Tacred
Revisited dataset, our SMS can bring at least 1.8 and up to 5.9 F1 score
improvement for the original encoder. Overall, different encoders with SMS all
can obtain remarkable improvement on both datasets. This proves that our SMS
really captures effective features from input sentence representations, which
can not get directly from the representations. Compared with models which
employ pre-trained models without external knowledge (i.e. training data or
knowledge graph) in block 2, pretrained models with our SMS in block 4 overall
perform better and SpanBERT-large+SMS achieve new state-of-the art results on
both datasets. In addition, we can see that the performance of SpanBERT-
large+SMS is better than MTB, KnowBERT-W+W, and KEPLER in block 3 and is
competitive with K-Adapter and LUKE.

The increase of F1 score is more conspicuous on Tacred Revisited compared with
Tacred. This phenomenon is further evidence that existing models have neared
the upper limit of Tacred, which have many mislabeled examples. Besides, we
can see that models based on SpanBERT all have a pretty good performance. This
phenomenon proves the importance of segment level features.

####  3.4.2 Compare with LSTM-based models

To further evaluate the effectiveness of our method SMS, we specially combine
SMS with LSTM encoder. We can observe that our model also outperforms the
model with LSTM encoder in block 1. Dependency-based models with graph neural
networks (C-GCN and C-AGGCN) have a remarkable performance on Tacred and
models which focus on segments (SA-LSTM) have a better performance on Tacred
Revisited. This phenomenon means that directly modeling the segment level
feature can not effectively overcome the noise from mislabeled examples and
the introduction of graph structure with dependency trees can help models
tackle some influence from wrong examples in the dataset itself.

However, our LSTM+SMS can outperform them on both datasets due to our mention-
aware segment attention can alleviate influence from mislabeled entity pairs
via modeling entity mention level feature and hierarchical structure.

###  3.5 Results on SemEval 2010 Task 8

Models | SemEval
---|---
F1($\Delta F1$)
LSTM | 82.7
LSTM+Attention | 84.0
TRE | 87.1
BERT-base | 87.9
BERT-large | 88.8
SpanBERT-base | 88.2
SpanBERT-large | 89.4
R-BERT Wu and He (2019a) | 89.3
MTB † | 89.5
KnowBert-W+W ‡ | 89.1
LSTM+SMS | 86.8(+4.1)
BERT-base+SMS | 88.4(+0.5)
BERT-large+SMS | 89.8(+1.0)
SpanBERT-base+SMS | 88.5(+0.3)
SpanBERT-large+SMS | 90.3(+0.9)
Table 3: Results on SemEval 2010 task8. †means that the model employs external
knowledge to pre-train the model. ‡means the model employs a knowledge base.

We also evaluate our SMS with different encoders on SemEval 2010 Task 8
dataset and results are reported in Tab. 3. We can observe that our SMS still
brings remarkable improvement for different encoders, especially for LSTM
encoders. SpanBERT-large+SMS outperforms all compared to strong baselines.
Besides, SpanBERT-large+SMS can beat models with external knowledge due to
this dataset being simpler than Tacred which only has 9 relations and shorter
input sentences. These reasons reduce the gain from the introduction of
external knowledge.

We also can see that the improvement of LSTM with SMS is up to 4.1% F1 score.
We guess that pre-trained models contain a lot of semantic information from
pre-training data which is similar to features from our SMS. However, LSTM
only captures features from the plain texts and can achieve more improvement
from our proposed SMS.

##  4 Discussion

###  4.1 Ablation Study

| F1($\Delta F1$)
---|---
SpanBERT-large | 78.0
\+ sentence level | 78.6(+0.6)
\+ mention level | 78.8(+0.8)
\+ segment level | 79.4(+1.4)
\+ all | 79.8(+1.8)
Table 4: Ablation study on Tacred Revisited test set.

To evaluate the contribution of each component of our SMS, we do an ablation
study and results are shown in Tab. 4. We can observe that segment level
features contribute the most for the F1 score, which are extracted by the
mention-aware segment attention. This means the hierarchical structure between
entity mention level and segment level feature really play a vital role for
relation prediction. In the future works, segment features need more
attention. We also can see that all three granularity features influence the
performance obviously. This proves the capture of these three granularity
features are proper for relation extraction tasks.

###  4.2 Analysis with N-gram Segments

Figure 3: Performance on Tacred Revisited test set with different n-gram
segments features. Number $n$ in x-axis means the model use $n$-gram segment
features.

We show the performance on the Tacred Revisited test set with different n-gram
segments features in Figure 3. Number $n$ in the x-axis means the model uses
$1-n$-gram segment features. We can observe that the model with 1,2,3-gram
segment features achieves the best performance. Longer segment features can
not bring improvement and may bring noise to the performance of the model. So
we employ 1,2,3-gram segment level features in our paper.

###  4.3 Case Study

Figure 4: Two examples which are sampled from Tacred Revisited test set. The
shade of the color represents how much attention is allocated. For the sake of
perception, we did not color words with very low attention values.

As shown in Figure 4, we visualized the attention of our SMS with two examples
which are sampled from Tacred test set. In the first example, our method
successfully pays more attention to entity mentions: “she”, “her”, “he”, and
“his”. All of them are key entity mentions for the predicted relation. We also
can observe that the mention-aware segment attention of our SMS can focus on
the core segment “her dad”, which is highly related to given entity pairs and
matches the predicted label “per:children”. From the second example, we can
see that the model learns additional information which is similar to target
relation. The model not only successfully pays attention on entity mention
“SUBJ-PER” and core segment “COO of”, but also captures similar entity mention
“Sally Strebel” and segment “CEO of” simultaneously. The case study proves
that the mention attention and mention-aware segment attention do capture
crucial entity mention level and segment level features.

##  5 Related Works

###  5.1 RE with Neural Networks

In recent years, neural networks have been large-scale used in relation
extraction (RE). Zeng et al. (2014); Nguyen and Grishman (2015); Wang et al.
(2016) employ convolutional neural networks (CNN) to extract lexical and
sentence level features for RE. Zhang and Wang (2015) employs bidirectional
recurrent neural networks (RNN) to learn long-term features to tackle long-
term relation problems in RE. And many models with different attention
mechanisms were proposed Zhou et al. (2016); Zhang et al. (2017); Xiao and Liu
(2016); Wang et al. (2016); Yu et al. (2019). Vu et al. (2016); Nayak and Ng
(2019) combine CNN and RNN to extract multi-types features from input
sentences. Recently, Verga et al. (2018); Liu et al. (2020) employ new neural
structure transformer to extract features for RE, which is based on self-
attention and is robust and powerful.

Different from previous sequence-based models, dependency-based models employ
dependency parsing of input sentences to capture non-local syntactic
relations. The use of dependency trees has been a trend in relation extraction
Xu et al. (2015); Cai et al. (2016); Miwa and Bansal (2016); Song et al.
(2018). Peng et al. (2017) split the dependency graph into two directed
graphs, then extended the tree LSTM model Tai et al. (2015) based on these two
graphs to learn the structure of syntax dependency. Zhang et al. (2018) first
introduced graph neural network Kipf and Welling (2017) (GNN) into RE model
for encoding featrues from dependency tree and proposed a pruning strategy to
remove unnecessary components of dependency tree. Guo et al. (2019) also
proposed a model with a soft-pruning approach that can automatically learn how
to selectively attend to the relevant sub-structures useful for relation
extraction.

###  5.2 RE with Pretrained Models

With the development of pre-trained language models Devlin et al. (2019), the
performance of relation extraction has been highly improved. After that, many
researches based on BERT were carried out. Most of these works employ pre-
trained language models in three ways for relation extraction: 1) design task-
related tasks in pre-training stage to improve prior pattern Zhang et al.
(2019); Joshi et al. (2020); Baldini Soares et al. (2019); Li and Tian (2020);
Peng et al. (2020); Yamada et al. (2020); 2) introduce external knowledge
(e.g. knowledge graph and wiki data) into fine-tuning or pre-training stages
Peters et al. (2019a); Baldini Soares et al. (2019); Wang et al. (2020b, a);
Yamada et al. (2020); 3) employ representation from pre-trained language
models and stack some neural structure over it Tao et al. (2019); Alt et al.
(2019); Wang et al. (2019); Wu and He (2019b); Shi and Lin (2019); Zhao et al.
(2019); Xue et al. (2020); Chen et al. (2020). There are also some special
methods with pre-trained language models Li et al. (2019); Zhao et al. (2020).
They convert relation classification tasks into machine reading comprehension
tasks. However, most of them is time-consuming or resource-consuming due to
the require of external knowledge and the pre-train stage.

##  6 Conclusion and Limitations

In this paper, we analyze previous typical works and empirically focus on
three granularity features: entity mention level, segment level and sentence
level. Based on the hierarchical structure between entity mention level and
segment level feature, we propose a multi-granularity hierarchical feature
extractor for relation extraction, which does not need any external knowledge
or tools. We evaluate our method with different encoders and results on three
public benchmarks show that our method can bring outstanding improvement for
them.

The structure of our model make it not easy to apply on multi-relation
extraction tasks. In the future, we will focus on how to extend our method to
longer input tasks and multi-relation extraction tasks (e.g. Document Level
Relation Extraction). Besides, we will also investigate what makes graph
structure effective in relation extraction tasks and why our method can obtain
better results than them.

## Acknowledgements

This work was supported in part by the National Natural Science Foundation of
China (Grant Nos.U1636211, 61672081,61370126), the 2020 Tencent Wechat Rhino-
Bird Focused Research Program, and the Fund of the State Key Laboratory of
Software Development Environment (Grant No. SKLSDE-2021ZX-18).

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# Hysteresis and return point memory in the random field Blume Capel model

B.E. Aldrin<EMAIL_ADDRESS>School of Physical Sciences, National
Institute of Science Education and Research, Bhubaneswar, P.O. Jatni, Khurda,
Odisha, India 752050  Homi Bhabha National Institute, Training School Complex,
Anushakti Nagar, Mumbai 400094, India  Abdul Khaleque<EMAIL_ADDRESS>Department of Physics, Bidhan Chandra College, University of Calcutta,
Kolkata, India  Sumedha<EMAIL_ADDRESS>School of Physical Sciences,
National Institute of Science Education and Research, Bhubaneswar, P.O. Jatni,
Khurda, Odisha, India 752050  Homi Bhabha National Institute, Training School
Complex, Anushakti Nagar, Mumbai 400094, India

###### Abstract

We study the zero temperature steady state of the random field Blume Capel
model with spin-flip Glauber dynamics on a random regular graph. The
magnetization $m$ as a function of the external field $H$ is observed to have
double hysteresis loops with a return point memory. We also solve the model on
a Bethe lattice in the approximation that the spin relaxation dynamics is
abelian and find good agreement between simulations on random regular graphs
and Bethe lattice calculations for negative values of $H$.

##  I Introduction

In experiments, it is often seen that a piece of magnet exposed to an
increasing external magnetic field magnetises in a series of jumps or
avalanches. This gives rise to a magnetization that lags behind the external
magnetic field and shows a history dependence. This effect is known as
hysteresis [1]. The magnetic signal produced on changing the external field
known as the Barkhausen noise, is highly reproducible [2]. Hysteresis is also
seen in many other systems (see [3] for a recent review) like martensites [4],
DNA unzipping [5], granular matter [6, 7], output vs input in economics [8]
and perception in psychology [9].

Systems with multiple metastable states with energy barriers higher than the
energy of the thermal fluctuations typically exhibit hysteresis on the
application of an external force. Spin systems with quenched disorder are
examples of systems with multiple metastable states. Sethna et al [10] studied
the zero temperature random-field Ising model (RFIM) and found that the model
exhibits hysteresis and a return point memory (RPM). The zero temperature RFIM
has been very useful in understanding the phenomenon of hysteresis and RPM.
The model has hence been studied extensively and continues to be of
interest[2, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]. RPM can be
described as the system’s ability to return to a previous state when the
direction of the applied magnetic field is reversed. This is important as it
allows magnetic tapes to be rerecorded and gives shape memory to shape memory
alloys [3]. The existence of RPM for zero temperature RFIM steady state with
spin-flip Glauber dynamics [26] was proved using the no passing property [10,
25].

The steady state magnetization for the RFIM with zero temperature Glauber
spin-flip dynamics exhibits a single hysteresis loop as the external field is
varied. In experiments, one often observes multiple hysteresis loops in
systems like ferroelectrics [27, 28], antiferroelectrics [29] and martensites
[30]. A connection between multiple hysteresis and higher spin molecules was
reported in experiments involving $Mn_{12}$ [31]. In most of these systems,
like the thermoelastic martensites, RPM is also seen in experiments [4].
Numerical studies involving higher spins and continuous spins with random
fields have also reported multiple hysteresis loops [32, 34, 33].

In this paper we study the hysteresis response of the zero temperature
spin-$1$ random field Blume Capel model (RFBCM) and find that the
magnetization exhibits double hysteresis (see Fig. 1) with RPM. The model is
defined by the Hamiltonian:

$\displaystyle H$ $\displaystyle=$
$\displaystyle-\sum_{<ij>}s_{i}s_{j}-\sum_{i}h_{i}s_{i}-H\sum_{i}s_{i}+\Delta\sum_{i}s_{i}^{2}$
(1)

Here $s_{i}$ are the spin-$1$ variables that take values $\pm 1$ and $0$. The
index $i$ and $j$ run over all the lattice sites. $\Delta$ is the crystal
field strength : a positive value of $\Delta$ favours the $0$ spin state,
while a negative value favours the $\pm 1$ spin states. $H$ is the external
magnetic field and $h_{i}$ is the quenched local random field, drawn from a
continuous Gaussian probability distribution
$\phi(h)=\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-h^{2}/{2\sigma^{2}}}$, with
$\sigma$ the standard deviation of the distribution. At $T=0$ in the absence
of the random field, the model has a first order transition from an ordered
state to a disordered state as a function of $\Delta$. In the presence of the
random Gaussian field the equilibrium phase diagram of the model on a fully
connected graph has been studied recently [35]. The phase diagram in the
$\Delta-\sigma$ plane has a line of critical transition ending at a
tricritical point. The zero temperature phase diagram of the model on random
regular graphs and finite dimensional lattices has not been studied and
numerical studies similar to that of RFIM will be useful for this model as
well [36].

We study the $T=0$ RFBCM numerically using the zero temperature Glauber spin-
flip dynamics on a random regular graph. At $T=0$, the Glauber spin-flip
dynamics lacks detailed balance and we get a non-equilibirum steady state in
which the magnetization shows hysteresis. It is useful to study models on a
random regular graph as they take correlations between neighbouring sites into
account. The nature of the steady state can be different on a random regular
graph than on a fully connected graph, in the presence of disorder. For
example, zero temperature RFIM on a fully connected graph shows hysteresis
only for $\sigma$ less than a critical value, whereas in the case of random
regular graphs, hysteresis is seen for all values of $\sigma$ [10, 11].

We find that even though the dynamics is not abelian [12] for RFBCM, zero
temperature steady state of the model exhibits RPM on a random regular graph.
We solve the model on a Bethe lattice by extending the method used to study
the RFIM [11]. The hysteresis plots obtained on the Bethe lattice match with
the plots for the same coordination number on a random regular graph for
negative values of $H$ and do not match for positive values of $H$. Since the
hysteresis loops on a random regular graph for RFBCM for positive and negative
$H$ are identical on reversing the sign, our Bethe lattice calculations
provide an approximate solution of the problem on a random regular graph. We
also find that the magnetization undergoes a transition from first order
hysteresis to second order hysteresis for RFBCM on random regular graphs with
coordination number $\geq 3$ at a value of $\sigma$ that changes with
$\Delta$. This is in contrast to the behaviour of RFIM where first order
hysteresis loops are seen only for coordination number $\geq 4$ [11].

The plan of the paper is as follows: In Sec. II we study hysteresis on random
regular graphs and show the RPM property of the hysteresis curves. In Sec. III
we solve the model on the Bethe lattice and compare these results with the
simulation results on a random regular graph. We summarise in Sec. IV.

##  II Hysteresis on a random regular graph

We study the zero temperature non-equilibrium steady state reached when the
magnetic field is increased adiabatically from minus infinity to plus infinity
on a random regular graph of coordination number $z$. The quenched local
random field $h_{i}$ at each site is drawn from a Gaussian distribution
$\phi(h_{i})$ with standard deviation $\sigma$. To study hysteresis start with
an initial state in which the external field $H$ is large and negative, such
that the steady state of the system has all spins in $-1$ state. The field is
increased in steps of $\delta H=0.01$. After each increment in $H$, the system
relaxes via zero temperature single spin-flip Glauber dynamics till a steady
state is reached (see Appendix A for details). In zero temperature Glauber
spin-flip dynamics, a spin changes its state if the change in energy
($\delta_{E}$) due to the spin-flip is negative. Since there are two possible
states to which a spin can flip, the one which lowers the energy more is
chosen. The whole lattice is updated till there are no flippable spins. The
change in energy depends on the initial and final spin state. There are three
possibilities for a spin ($s$) at site $i$ to flip:

  * •

$s=\pm 1$ to $\mp 1$ : $\delta_{E}=2s(\sum_{j\in\mathcal{N}(i)}s_{j}+h_{i}+H)$

  * •

$s=\pm 1$ to $0$ :
$\delta_{E}=s\left(\sum_{j\in\mathcal{N}(i)}s_{j}+h_{i}+H\right)-\Delta$

  * •

$0$ to $s=\pm 1$ :
$\delta_{E}=-s(\sum_{j\in\mathcal{N}(i)}s_{j}+h_{i}+H)+\Delta$

here $\mathcal{N}(i)$ denotes the set of neighbours of the site $i$.

We considered many different network configurations and found that for large
system sizes the change in magnetization from one realization of disorder to
the other is not noticeable (see Appendix A). Simulations were done with
$N=10^{5}$, $N$ being the system size. The magnetization is simply the sum of
all spin values in the steady state divided by the system size $N$.

###  II.1 Hysteresis

Figure 1: Hysteresis response of magnetization ($m$) for random regular graph
with coordination number $z=3$ and $z=4$ is plotted for two different $\sigma$
values as a function of external field ($H$) for a typical realization of the
disorder. In Fig. 1a, the red (light gray) line of plusses and the blue (dark
gray) line of circles correspond to $\sigma=0.15$ and $1.0$ respectively. In
Fig. 1b, the red (light gray) and the blue (dark gray) lines correspond to
$\sigma=0.5$ and $1.5$ respectively. In both cases the hysteresis loop shows
discontinuity for small $\sigma$ and transitions to continuous hysteresis loop
for large $\sigma$. The value of $\sigma$ at which the transition occurs
increases with $z$. For $z=3$ and $\Delta=3$, we find $\sigma_{c}=0.40(5)$ and
for $z=4$ and $\Delta=3$ we find $\sigma_{c}=0.9(2)$.

The magnetization in the steady state at the same value of $H$ is different
when the system is evolved starting with a very large negative external field
with all spins down than when the systems is evolved starting with a very
large positive external field with all spins up.

We find that the RFBCM shows two jumps in the magnetization and hence double
hysteresis for positive values of $\Delta$. In Fig. 1 the hysteresis curves
for $z=3$ and $z=4$ for different values of $\Delta$ and $\sigma$ are shown.
The magnetization jumps discontinuously for smaller $\sigma$ and smoothens as
$\sigma$ is increased. The magnetization undergoes a transition from being
discontinuous to continuous as $\sigma$ is increased for all $z\geq 3$.

For $\Delta\leq 0$, the $\pm 1$ spin states are favoured over the $0$ spin
state and the RFBCM has a behaviour similar to the RFIM with single hysteresis
loops. In Appendix B we give the magnetization plots of different values for
$z$, $\Delta$ and $\sigma$.

###  II.2 Return-Point Memory (RPM)

On changing the external field from a value $H_{1}$ at time $t=0$ to $H_{2}$
at time $t=t_{1}$ and backtracking from $H_{2}$ to $H_{1}$, if the
magnetization returns to its original value independent of the time $t_{1}$
and details of the evolution of the external field, then the system has RPM.
We find magnetization in the case of RFBCM exhibits RPM for all values of
$\Delta$ and $\sigma$. For $z=4,\Delta=3$ and $\sigma=1$ it is shown in Fig.
2.

RPM is widely observed in experiments. In the case of RFIM, the presence of
RPM was proven using the existence of no passing property [25]. No passing
property for spin systems implies that if we take two configurations
$C^{\alpha}$ and $C^{\beta}$ such that the spins in the two configurations are
partially ordered ( for all the spins in the two configurations
$s_{i}^{\alpha}\geq s_{i}^{\beta}$ ), then the partial ordering is preserved
under the application of the external field as long as the field
$H^{\alpha}(t)$ applied to $C^{\alpha}$ is always greater than or equal to the
field $H^{\beta}(t)$ applied to $C^{\beta}$. The no passing property implies
abelian dynamics for zero temperature RFIM. The abelian dynamics for RFIM
means that if we start from a configuration with all spins in the $-1$ state
and external field $-\infty$ and increase the field to a finite value $H$ in
one step then the order in which unstable spins are relaxed does not matter
[11, 37].

No passing property holds also for the zero temperature RFBCM. The dynamics is
not abelian for RFBCM i.e, the order of flipping of the unstable spins now can
change the configuration attained. Although if we combine the $0$ and $+1$
states into one state $X$, we can think of the RFBCM as an effective two state
system with abelian dynamics. In Section III, we make use of this observation
to find the magnetization of the model on a Bethe lattice.

Figure 2: Double hysteresis loops for zero temperature RFBCM between
magnetization ($m$) and external field ($H$), showing RPM. The two hysteresis
loops are identical and RPM exists between any two different values of $H$.

###  II.3 Coercive field and remanent magnetization

The coercive field $H^{*}$ is defined as the value of $H$ beyond which the
negative $m$ solution disappears [26]. Thus, in simulations while evolving the
system, increasing the external field from a large negative value, the
coercive field is taken to be the field $H^{*}$ for which the magnetization
$m$ of the system becomes non-negative for the first time. As shown in Fig. 3,
at small $\sigma$, $H^{*}$ decays rapidly. This is because the hysteresis loop
is first order in this region and moves left on increasing $\sigma$ as more
and more spins take the $0$ value. However as $\sigma$ increases further,
$H^{*}$ increases with increasing $\sigma$. This change in slope is observed
to occur before $\sigma_{c}$.

Another useful property is the remanent magnetization ($m_{R}$) which is the
steady state value of the magnetization when starting from the initial state
with all spins up or down, the external field $H$ is made zero [26]. For RFBCM
at $\sigma=0$ the value of $m_{R}$ is $z-\Delta$. On increasing $\sigma$, it
first decreases as the first order hysteresis loop moves towards the left
along the $H$ axis. It goes to zero $0$ at a value of $\sigma$ which is well
before the transition from the first order hysteresis to the continuous
hysteresis. It continues to be zero for $\sigma<\sigma_{c}$ and increases with
$\sigma$ beyond that (see Fig. 4).

Figure 3: Coercive field ($H^{*}$) for $z=4$, $\Delta=3$ as a function of
$\sigma$. Figure 4: Remanent magnetization ($m_{R}$) plot for $z=4$,
$\Delta=3$. Here the blue (lower) line, $m_{FR}$ and the red (upper) line,
$m_{BR}$ denote the remanent magnetization $m_{R}$ curves obtained from
forward and backward increments in H.

##  III Model on the Bethe lattice

For large system sizes, the loops in the random regular graph can be ignored
after which it is like a Bethe lattice of the same coordination number. A
Bethe lattice is the interior of a Cayley tree, such that the surface of the
tree does not influence the statistical averages. The zero temperature
hysteresis in RFIM was solved making use of the abelian property of the
dynamics and it was shown that there was an excellent match with the
simulations on a random regular graph [11].

Figure 5: Cayley tree with coordination number $z=3$ and height $4$.

In the case of the RFBCM that we study here, since the spins can flip to
either a $0$ state or $+1$ state on increasing $H$, the order of the spin-flip
can change the configuration achieved. The model exhibits RPM in spite of the
absence of the abelian property of the dynamics. The abelian property still
holds if we consider a two state system with $-1$ spin state and another state
$X$, which can either be $0$ or $+1$. We hence expect the dynamics to be
almost abelian for large negative $H$. We calculate the magnetization for the
model on a Bethe lattice of coordination number $z$ in this section under this
assumption and find good agreement with the simulations.

Consider a Cayley tree where all non-boundary sites have coordination number
$z$ and the boundary sites have coordination number $1$. Different levels of
the tree are labelled by $r$ with $r=0$ for the root and $r=n$ for the
boundary sites. Fig. 5 shows a Cayley tree with $n=4$ levels for $z=3$. Start
with $H=-\infty$, with all spins in the $-1$ state. Now increase the field to
$H$ in a single step and relax the spins starting with the spins in the level
$r=n$, keeping all the spins in other levels frozen in $-1$ state. Next relax
the spins in level $r=n-1$, keeping the spins in levels below $n-1$ fixed in
$-1$ state. Continue this process till all the spins on the tree are relaxed.

Since a spin can flip either to a $0$ state or a $+1$ state, we define
probability $P_{r}$ ($Q_{r}$) to be the probability that a spin at level $r$
will flip to $+1$ ($0$) state given that the spins in levels $r+1$ and above
have already been relaxed and the spins in levels $r-1$ and below are all in
$-1$ state. The spin-flip probabilities of different spins in the same level
are independent of each other and the recursion relations for $P_{r}$ and
$Q_{r}$ are

$\displaystyle
P_{r}=\sum_{s_{0}=0}^{(z-1)}\sum_{s_{1}=0}^{(z-1)-s_{0}}\frac{(z-1)!}{s_{0}!s_{1}!((z-1)-s_{1}-s_{0})!}Q_{r+1}^{s_{0}}$
$\displaystyle
P_{r+1}^{s_{1}}(1-Q_{r+1}-P_{r+1})^{(z-1)-s_{0}-s_{1}}p_{s_{0}s_{1}}$ (2)
$\displaystyle
Q_{r}=\sum_{s_{0}=0}^{(z-1)}\sum_{s_{1}=0}^{(z-1)-s_{0}}\frac{(z-1)!}{s_{0}!s_{1}!((z-1)-s_{1}-s_{0})!}Q_{r+1}^{s_{0}}$
$\displaystyle
P_{r+1}^{s_{1}}(1-Q_{r+1}-P_{r+1})^{(z-1)-s_{0}-s_{1}}q_{s_{0}s_{1}}$ (3)

where $p_{s_{0}s_{1}}$($q_{s_{0}s_{1}}$) is the probability that a $-1$ spin
with $s_{1}$ descendants in $+1$ state and $s_{0}$ in $0$ state will flip to
$+1$($0$) state. They are given by

$\displaystyle
p_{s_{0}s_{1}}=\int_{-H+(z-2s_{1}-s_{0})+\Delta}^{\infty}\phi(h_{i})dh_{i}.$
(4) $\displaystyle
q_{s_{0}s_{1}}=\int_{-H+(z-2s_{1}-s_{0})-\Delta}^{-H+(z-2s_{1}-s_{0})+\Delta}\phi(h_{i})dh_{i}.$
(5)

As $n\rightarrow\infty$, $P_{r}$ and $Q_{r}$ take their fixed point values
$P^{*}$ and $Q^{*}$ respectively, which have recursions given by

$\displaystyle
P^{*}=\sum_{s_{0}=0}^{(z-1)}\sum_{s_{1}=0}^{(z-1)-s_{0}}\frac{(z-1)!}{s_{0}!s_{1}!((z-1)-s_{1}-s_{0})!}(Q^{*})^{s_{0}}$
$\displaystyle(P^{*})^{s_{1}}(1-Q^{*}-P^{*})^{(z-1)-s_{0}-s_{1}}p_{s_{0}s_{1}}$
(6) $\displaystyle
Q^{*}=\sum_{s_{0}=0}^{(z-1)}\sum_{s_{1}=0}^{(z-1)-s_{0}}\frac{(z-1)!}{s_{0}!s_{1}!((z-1)-s_{1}-s_{0})!}(Q^{*})^{s_{0}}$
$\displaystyle(P^{*})^{s_{1}}(1-Q^{*}-P^{*})^{(z-1)-s_{0}-s_{1}}q_{s_{0}s_{1}}$
(7)

The value $P_{C}$ and $Q_{C}$ at the root node $C$ as $n\rightarrow\infty$ is
then given by

$\displaystyle
P_{C}=\sum_{s_{0}=0}^{z}\sum_{s_{1}=0}^{z-s_{0}}\frac{z!}{s_{0}!s_{1}!(z-s_{1}-s_{0})!}(Q^{*})^{s_{0}}$
$\displaystyle(P^{*})^{s_{1}}(1-Q^{*}-P^{*})^{z-s_{0}-s_{1}}p_{s_{0}s_{1}}$
(8) $\displaystyle
Q_{C}=\sum_{s_{0}=0}^{z}\sum_{s_{1}=0}^{z-s_{0}}\frac{z!}{s_{0}!s_{1}!(z-s_{1}-s_{0})!}(Q^{*})^{s_{0}}$
$\displaystyle(P^{*})^{s_{1}}(1-Q^{*}-P^{*})^{z-s_{0}-s_{1}}q_{s_{0}s_{1}}$
(9)

The probabilities $P_{C}$ and $Q_{C}$ are the values deep inside the tree far
away from the surface and are the estimate of the fraction of the $+1$ and $0$
spins respectively on a Bethe lattice. Magnetization per spin on a Bethe
lattice is given by $m=2P_{C}+Q_{C}-1$.

###  III.1 Results

The solution obtained above matches with the simulations on the random regular
graph of the same coordination number for negative values of the external
field. In Fig. 6 we plot the magnetization $m$ obtained via the above
calculation with the simulation results on a random regular graph for $z=3$,
$\Delta=3$, $\sigma=1$ and for $z=4$, $\Delta=3$, $\sigma=1.5$. The behaviour
is similar for other sets of ($z$, $\Delta$, $\sigma$) as well (see Appendix
C). We also plot the sum of $P_{C}$ and $Q_{C}$ along with their values from
simulations on a random regular graph of the same coordination number in Fig.
7.

Figure 6: Magnetization $m$ on a Bethe lattice (BL) (the blue (light gray)
line of circles) as a function of external magnetic field $H$ for coordination
numbers $z=3,4$ is compared with the value from simulations on a random
regular graph (rrg) (the red (dark gray) line of plusses) for same value of
$z,\Delta,\sigma$.

Figure 7: $P_{C}+Q_{C}$ as a function of external magnetic field $H$ from the
Bethe lattice (BL) (the blue (light gray) line of circles) calculations is
compared with the sum of the probabilities of $+1$ spin state and $0$ spin
state from the simulations on random regular graphs (rrg) (the red (dark gray)
line of plusses). The two plots overlap showing a good match between the
analytical calculation and simulations.

Figure 8: $P_{C}$ and $Q_{C}$ as a function of external magnetic field $H$ for
$z=3$ and $z=4$ is shown. $Q_{C}$ is overestimated and $P_{C}$ is
underestimated for positive $H$ in the Bethe lattice (BL) (the blue (light
gray) line of circles calculations when compared with the value on a random
regular graph (rrg) (the red (dark gray) line of plusses).

As shown in Fig. 6, $m$ matches with the simulated data for $H<0$ but deviates
for positive $H$. For spin-$1$, we find an underestimation for $P_{C}$ and an
overestimation for $Q_{C}$, as $H$ increases. In principle once a spin at
level $r$ flips, it can change the configuration of the spins at level $r+1$,
which has been ignored in the calculations on the Bethe lattice. This though
does not change the value of the probabilities at node $C$ for RFIM as a
change in $r+1$ cannot further change the spin at level $r$ (it is already in
$+1$ state). But the same is not true for RFBCM model. For example, let us
assume that a spin at level $r$ attains a $0$ value due to its descendants and
in turn makes its descendants flippable and hence they change their state. It
is now possible that its descendant spins in level $r+1$ produce enough
interaction field at the spin that now the same spin at level $r$ might prefer
to be in $+1$ state. Ignoring this results in the underestimation of $P_{C}$
and overestimation of $Q_{C}$ as $H$ is increased (see Fig. 8). The sum
$P_{C}+Q_{C}$ is correctly calculated in the Bethe lattice calculation. In
Fig. 7 the plot of $P_{C}+Q_{C}$ on a Bethe lattice and the one obtained in
simulations on a random regular graph overlap for all values of $H$.

A solution on the Bethe lattice under the abelian assumption fairly accurately
calculates the lower hysteresis loop. Since the $m$ vs $H$ curves for random
regular graphs as shown in Fig. 1 for positive and negative $H$ can be
superimposed on each other by reversing the sign, the solution obtained in
this section gives the solution of the problem on a Bethe lattice. On the
Bethe lattice for $z\geq 3$, $P^{*}$ and $Q^{*}$ become multivalued when $H$
is near the transition for $\sigma<\sigma_{c}$. The hysteresis is thus
discontinuous for $\sigma<\sigma_{c}$ and continuous for $\sigma>\sigma_{c}$
$\forall z\geq 3$ also on a Bethe lattice. The value of $\sigma_{c}$ depends
on $z$ and $\Delta$. The value of $\sigma_{c}$ calculated from the Bethe
lattice calculations is found to be consistent with the estimate from the
simulations on a random regular graph. We tabulate the value of $\sigma_{c}$
from simulations and the Bethe lattice calculations for a few values of
$\Delta$ for $z=4$ in Table.1.

$z$ | $\Delta$ | $\sigma_{c}(simulations)$ | $\sigma_{c}(Bethe\ lattice)$
---|---|---|---
4 | -1 | 1.8(2) | 1.9(1)
4 | 1 | 1.8(2) | 1.9(1)
4 | 3 | 0.9(2) | 1.1(1)
4 | 5 | 0.9(2) | 1.1(1)
Table 1: Critical value of $\sigma$ at which the hysteresis plot become
continuous, as estimated in the simulations and calculations on the Bethe
lattice.

##  IV Discussions

The zero temperature RFIM has been studied extensively as it has been
successful in explaining the mechanism of hysteresis for many experimental
systems [2]. In this paper, we studied the zero temperature spin-$1$ RFBCM
numerically on a random regular graph and extended the earlier calculation of
the magnetization for RFIM on a Bethe lattice to RFBCM. We showed that the
Bethe lattice calculations match with the simulation results on a random
regular graph for negative $H$ in spite of the absence of the abelian property
of the Glauber dynamics for RFBCM.

It would be interesting to expand this study to other higher spin
ferromagnetic random field models. In particular zero temperature spin-$1$
random field Blume Emery Griffiths model (RFBEGM), which differs from RFBCM by
the presence of an extra bi-quadratic interaction [38]. The zero temperature
RFBEGM was numerically studied on a square lattice in [32] as a model for
martensite transitions by considering the $0$ and $\pm 1$ spin states as
austenite and two martensite states respectively. They observed that the model
exhibits double hysteresis without RPM [32]. In experiments, RPM is almost
always observed in thermoelastic martensite transitions [4]. In this paper, we
show that a spin-$1$ random field model can also exhibit RPM. This suggests
that the presence of the biquadratic term results in the loss of RPM property
for a spin-$1$ ferromagnet. In general, the conditions under which a system
exhibit RPM are not yet fully understood [39, 40]. Critical properties of the
zero temperature non equilibrium steady state of spin-$1$ models like RFBCM
and RFBEGM is largely unexplored and it would be interesting to study these
models in different dimensions.

##  V Acknowledgements

We thank Deepak Dhar for discussions. A.K thanks NISER, Bhubaneswar for local
hospitality when the part of this work was done.

## Appendix A

###  A.1 Thermalization time

Thermalization time $\tau$ is defined as the number of Monte Carlo steps taken
by the system to reach a steady state, for a fixed value of the parameters
like the external field. One sweep of the entire system is taken as the one
Monte Carlo step. Hence $\tau$ for a given $H$ is the total number of times we
run through the whole lattice looking for possible spin flips, until we reach
a spin configuration that yields no spin flips at any of the lattice sites. We
plot $\tau$ as a function of the external field $H$ against the magnetization
plots for $z=4$, $\Delta=3$ and for different values of $\sigma=0.5,1.5,3$ in
FIG. 9. The $\tau$ as expected increases near the transition.

Figure 9:  (Appendix A.1) Thermalization time ($\tau$) plots are shown against
magnetization ($m$) plots for z = 4, $\Delta=3$ for different values of
$\sigma=0.5,1.5,3$.

Figure 10: (Appendix A.2) The magnetization plots of RFBCM obtained for two
different random field realizations (RFR1 and RFR2) are indistinguishable at
the scale of the graph, as shown for two different values of $\sigma=0.5,1.5$.

###  A.2 Comparing magnetization plots of different random field realizations

To show that for large system sizes the change in magnetization from one
realization of disorder to the other is not noticeable, we plot the
magnetization for $z=4$, $\Delta=3$, for two different random field
realizations for $\sigma=0.5$ and $1.5$. For both of the values of $\sigma$,
the plots obtained from different random field realizations (RFR1 and RFR2)
are indistinguishable at the scale of the graph (as shown in FIG.10).

###  A.3 Magnetization plots of different increments in the field

The magnetization for $z=4$, $\Delta=3$ and $\sigma=1.5$ obtained for
different values of the increments in the external field ($\delta H$
=$10^{-2}$, $10^{-3}$, $10^{-4}$, $10^{-5}$, $10^{-6}$) are plotted in Fig.
11. All the plots are indistinguishable. Hence we have taken $\delta
H=10^{-2}$ in our simulations.

Figure 11: (Appendix A.3) The magnetization plots of RFBCM obtained for
different values of the increments in the external field ($10^{-2}$,
$10^{-3}$, $10^{-4}$, $10^{-5}$, $10^{-6}$) that are indistinguishable at the
scale of the graph.

##  Appendix B Hysteresis plots for various values of $z$,$\Delta$ and
$\sigma$

We plot the magnetization plots of RFBCM for different values of the
parameters ($\Delta$ and $\sigma$) for $z=3$ (Fig. 12) and $z=4$ (FIG. 13).

Figure 12: (Appendix B) The magnetization for different values of the
parameters ($\Delta=-1,1,3$ and $\sigma=0.15,0.5,1$) for $z=3$ are shown
above.

Figure 13: (Appendix B) The magnetization for different values of the
parameters ($\Delta=-1,1,3$ and $\sigma=0.5,1.5,3$) for $z=4$ are shown above.

##  Appendix C Comparison of the magnetization from Bethe calculations with
the simulations

In Fig.14, we compare the magnetization plots obtained from Bethe lattice
calculations with that obtained from the simulations for $z=4$, $\Delta=3$ for
$\sigma=0.7,0.8,0.9,1,2$ and $3$. In the negative $H$ region there is a good
match between the simulations and the Bethe lattice calulcations. As expected,
the mismatch increases with increasing $\sigma$.

Figure 14: (Appendix C) Magnetization from Bethe lattice (BL) calculations is
compared with that from simulations for forward magnetization curve on a
random regular graph (rrg) for $z=4$ and $\Delta=3$ for various values of,
$\sigma=0.7,0.8,0.9,1,2$ and $3$.

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