Extremal Dependence and Community-Structured Risk Propagation in Complex Social Information Networks

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Extremal Dependence and Community-Structured Risk Propagation in Complex Social Information Networks

读者定位：这份笔记按“没有数理基础也能读懂”的方式写。先给直觉，再给专业表达，再解释公式。英文原文来自用户提供 PDF 的文本层抽取；中文部分是面向学习的直译与解释合并稿。

0. 一句话读懂

这篇论文想解决的问题是：当社交网络里出现极端舆论爆发时，哪些用户和社区会在“极端状态”下一起异常活跃，以及应该优先盯住哪些社区之间的传播接口。

1. 论文在做什么

• 普通相关系数看的是平均情况下两个用户是否一起变化。
• 但危机传播往往只在少数极端时刻发生，所以作者改看 tail regime。
• 作者用 TPDM 把极端同步关系变成边权网络。
• 在这个网络上计算节点风险、发现社区，再用 minimum cut 找候选干预接口。
• 论文实验是 synthetic network 概念验证，不能直接当真实平台结论。

2. 术语表

3. 每一个公式的直白解释

4. 概念之间的关系

flowchart TD

  A["社交网络用户信号"] --> B["多维风险向量 X_i(t)"]

  B --> C["极端状态 R_uv(t) 超过高分位数"]

  C --> D["TPDM: 极端同步强度"]

  D --> E["极端依赖加权网络"]

  E --> F["节点风险 R_i"]

  E --> G["社区发现"]

  F --> H["高风险节点排序"]

  G --> I["社区图"]

  I --> J["minimum cut 候选干预接口"]

5. 中英对照精读稿

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Article

Extremal Dependence and Community-Structured Risk

Propagation in Complex Social Information Networks

Liang Wei 1 , Hanzhi Wang 2, and Yi Sun 3

1 Shanghai Data Group Co., Ltd., No.55, Wangda Rd., Shanghai, China; dreamfollower@yeah.net

2 Shanghai Institute of Early Childhood Education, Shanghai Normal University, No.100 Guilin Rd., Shanghai,

China; 1000533247@smail.shnu.edu.cn

3 Research Institute of Intelligent Complex Systems, Fudan University, Shanghai, China;

* Correspondence: 1000533247@smail.shnu.edu.cn

Abstract

Extreme opinion propagation in social information networks often appears as a low-

frequency but high-impact process, in which abnormal activity becomes synchronized

across structurally related users or communities during crisis periods. Conventional

correlation-based methods mainly describe average co-movement and may therefore miss

dependence patterns that emerge only in the tail regime. To address this issue, this paper

proposes a community-structured extremal dependence framework for social opinion

propagation risk analysis. A tail pairwise dependence matrix (TPDM) is used to construct

a weighted extremal dependence network, on which node-level risk scoring, community

detection, and community-level intervention analysis are performed. The proposed risk

score integrates degree centrality, betweenness centrality, tail exposure, and community

embedding strength, while the intervention component is formulated as a minimum-cut

problem on the induced community graph. The framework is evaluated on a controlled

synthetic social discussion network with 100 nodes. The experiment is intended as a

methodological proof of concept rather than as a real-platform empirical validation. The

results show that the TPDM-based network produces a structured representation with

two dominant coupled communities, several peripheral singleton nodes, concentrated

high-risk nodes, and one principal source–target interface in the community graph. These

findings indicate that extremal dependence can provide a useful representation of candidate

risk-coupling structures under the synthetic setting. However, the inferred edges should

not be interpreted as causal propagation paths, and the minimum-cut result should be

understood as a candidate intervention interface rather than as a guarantee of complete

diffusion blockage. Future work should validate the framework on real social media traces,

incorporate temporal causal information, and examine robustness under multi-channel

diffusion and adaptive user behavior.

Keywords: social network; opinion propagation; tail pairwise dependence matrix (TPDM);

extremal dependence; community detection; minimum cut; risk assessment

Published: 1. Introduction

Copyright: © 2026 by the authors.

With the rapid expansion of online social platforms, public opinion dissemination

Submitted to Mathematics for possible

has become increasingly decentralized, dynamic, and structurally heterogeneous. Users

open access publication under the 30

terms and conditions of the Creative are connected through replying, reposting, mentioning, and co-participation behaviors, 31

Commons Attribution (CC BY) license. thereby forming complex interaction networks in which local fluctuations may evolve 32

中文:

本页给出论文题目、作者、摘要和关键词。核心意思是：社交网络里的极端舆论传播不是天天发生，但一旦发生影响很大；普通相关系数只能看平均情况下谁和谁一起动，容易漏掉危机时刻才出现的尾部联动。因此作者提出 TPDM 极端依赖框架，把用户之间在极端状态下的同步程度变成一张加权网络，再做节点风险排序、社区发现和社区层面的干预接口识别。作者明确说实验只是 100 个节点的合成网络概念验证，不是真实平台验证。

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into large-scale propagation cascades [1,2]. In particular, extreme opinion propagation

often appears in a low-frequency but high-impact form, characterized by sudden bursts

of synchronized activity, emotional amplification, and cross-community diffusion. Such

processes are difficult to explain from isolated user activity alone, because their occurrence

is strongly shaped by structural dependence and collective interaction patterns.

Recent empirical studies on misinformation, rumors, and online fact-checking further

highlight the practical relevance of this problem. False and misleading information can

spread rapidly through social media platforms, and its diffusion is often shaped by user

interaction structures, echo chambers, topic communities, and platform-level interventions

[3,4]. Public datasets such as FakeNewsNet, CREDBANK, and PHEME have also shown

that misinformation research increasingly relies on social context, propagation traces,

temporal dynamics, and credibility annotations rather than on content features alone [5–7].

In addition, recent work on community-based fact-checking, including studies of X/Twitter

Community Notes, shows that intervention effectiveness depends not only on identifying

false content but also on when and where corrective signals enter the propagation network

[8,9]. These studies suggest that extreme opinion propagation should be examined as a

networked and temporally concentrated risk process.

Traditional methods for opinion-risk analysis mainly rely on descriptive statistics,

sentiment aggregation, supervised classification, or linear correlation analysis. While

these tools are useful for capturing average activity patterns, they are not designed to

characterize synchronized extreme behavior in the tail regime. In realistic social systems,

users or groups that appear weakly correlated under normal conditions may exhibit strong

co-fluctuation when extreme events occur, such as controversial incidents, emotionally

charged discussions, coordinated amplification, or sudden rumor cascades. Therefore, a

key methodological challenge is to quantify tail dependence between nodes and to further

understand how such dependence is organized and propagated through the network

structure.

From a complex systems perspective, network science has repeatedly shown that

diffusion robustness and vulnerability are determined not only by local node activity

but also by structural patterns such as hub dominance, bridge connectivity, and modular

organization [10,11]. Meanwhile, studies on community structure have demonstrated that

densely connected modules can serve both as local amplifiers and as mesoscopic units of

risk aggregation [12,13]. These observations suggest that a suitable risk model for extreme

opinion propagation should simultaneously account for both tail co-fluctuation behavior

and topology-driven transmission pathways.

To address this issue, this paper introduces the tail pairwise dependence matrix

(TPDM) into social opinion propagation analysis. TPDM was proposed as an analogue of

the covariance matrix for extremes under multivariate regular variation, enabling a positive

semi-definite summary of pairwise extremal dependence and supporting high-dimensional

analysis in the tail regime [14]. Compared with Pearson correlation, TPDM is tailored to

describe synchronized behavior under extreme conditions, which is precisely where high-

impact propagation events concentrate. Subsequent studies on extremal graphical modeling

have further shown that tail dependence structures can be represented as sparse graphs

and used to characterize dependence patterns under extreme regimes [15]. Motivated by

these advances, we use TPDM to capture synchronized tail-risk between users and then

map these extremal dependencies into a graph representation to support network-based

diagnostics.

In addition to pairwise extremal dependence, the organization of dependencies at the

mesoscopic level is essential for actionable intervention. Community structure analysis

provides a principled way to identify densely connected propagation groups, within which

中文:

本页继续引言。作者说明社交平台中的转发、回复、提及和共同参与会形成复杂网络。极端传播常表现为突发同步、情绪放大和跨社区扩散。已有谣言、错误信息和事实核查研究说明该问题有现实意义，但很多方法关注平均传播、单个用户活跃度或普通相关性，难以刻画只有在尾部才显现的极端共同波动。

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risk can spread rapidly, as well as bridge nodes and bridge edges that connect otherwise

separated communities and may serve as pathways for cross-community diffusion [12,13].

Accordingly, this paper combines TPDM-based network construction with community

detection to identify tightly coupled high-risk communities and bridge nodes prone to

amplifying extreme propagation. Furthermore, to move from risk identification to interven-

tion design, we aggregate the original network into a community graph and formulate a

minimum-cut optimization problem to derive least-cost blocking strategies between source

and target communities.

It should be emphasized that the present empirical analysis is based on a controlled

synthetic social discussion network. This design allows the proposed framework to be

examined under known structural conditions and avoids privacy and platform-access

restrictions associated with real social media data. However, it also limits the external

validity of the conclusions. Therefore, the empirical results should be interpreted as a

methodological proof of concept, and validation on real social media datasets remains an

important direction for future work.

The main contributions of this paper are as follows:

1. We introduce a TPDM-based graph construction framework for representing finite-

threshold extremal dependence among user-level risk vectors. The resulting network

is designed to capture synchronized tail behavior that may not be visible from average-

dependence measures alone.

2. We propose a composite node-level risk score that combines local connectivity, global

brokerage, tail exposure, and community embedding strength. This score is used as a

prioritization index for identifying structurally and extremally exposed nodes, rather

than as a causal probability of propagation.

3. We develop a community-level representation of extremal coupling and formulate a

minimum-cut problem on the induced community graph. The cut set is interpreted as

a candidate intervention interface for monitoring, attenuation, or resource allocation,

not as a literal removal of social relations.

4. We evaluate the framework on a controlled synthetic social discussion network. The

experiment examines whether TPDM-based network construction, node ranking,

community detection, and community-level cut analysis can operate as a coherent

analytical pipeline. The validation scope is explicitly limited to methodological

feasibility under synthetic conditions.

5. We discuss the major limitations of the framework, including the distinction between

dependence and causation, the absence of real-platform validation, threshold sensitiv-

ity, external common shocks, and adaptive multi-channel user behavior.

Rather than claiming universal superiority over existing correlation-based or topology-

based approaches, the proposed framework provides a complementary extremal-topological

representation for identifying candidate risk-coupling structures under rare but high-impact

conditions.

The rest of the paper is organized as follows. Section 2 introduces the preliminaries

and the proposed methodology. Section 3 presents the extremal-topological risk framework,

reports the regularity properties of the proposed risk score, and formulates the community-

level minimum-cut intervention model. Section 4 provides the empirical results on a

synthetic social discussion network. Section 5 discusses the main findings. Section 6

concludes the paper and suggests directions for future research.

2. Preliminaries and Methodology

This section details the proposed framework, which consists of five core components:

the tail pairwise dependence matrix (TPDM), network centrality indicators, community

中文:

本页收束引言并列出贡献。论文想把“极值理论”和“复杂网络”结合起来：先用 TPDM 表示有限样本中的极端同步关系，再把它变成图；再给每个节点算综合风险分数；再把社区之间的极端耦合抽象为社区图，用 minimum cut 找候选干预边界；最后在合成网络上演示。作者也强调边不能被解释为因果传播路径。

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structure analysis, an integrated node-level risk model, and a community-level intervention-

interface selection strategy. The overall goal is to quantify extremal co-fluctuation between

users and translate it into a graph representation that supports both node-level prioritization

and community-level intervention.

In a social discussion network, each node is associated with multiple propagation-

related signals, such as posting burst intensity, emotional extremeness, keyword burst level,

reply expansion rate, and cross-community interaction frequency. Therefore, we model

each node by a multivariate risk vector and focus on dependence behavior in the tail regime

where extreme propagation events concentrate.

2.1. Tail Pairwise Dependence Matrix (TPDM)

In social opinion propagation networks, it is crucial to understand whether two nodes

tend to exhibit extreme states simultaneously, because many high-impact propagation

events are coordinated or cascading rather than independent. For example, when a contro-

versial topic intensifies, users in different parts of the network may simultaneously exhibit

surges in posting frequency, emotion intensity, or cross-community interaction. The tail

pairwise dependence matrix (TPDM) provides a principled summary of such dependence

by focusing specifically on the tail of the joint distribution [14,15]. Unlike Pearson corre-

lation, which measures average linear association and may underestimate synchronized

extremes, TPDM targets dependence conditioned on large radial magnitude.

Let node u be associated with an m-dimensional risk vector

( )⊤

X (t) = X (t), X (t), . . . , X (t) ,

u u,1 u,2 u,m 152

where t = 1, . . . , T denotes the observation time index. The components of X (t) may repre-

u 153

sent propagation-related indicators such as posting burst intensity, sentiment extremeness,

keyword burst level, reply expansion rate, and cross-community interaction frequency.

For a pair of nodes (u, v), define the joint radial component as

||( )||

R uv (t) = || X u (t), X v (t) || 2 . (1) 157

The TPDM entry between the i-th component of node u and the j-th component of node v

is defined as

σ i ( j u,v) = x l → im ∞ E [ X R u u, v i · X R u v, v j | | | | R uv > x ] . (2) 1 1 6 5 0 9

This quantity measures the extent to which the two components tend to become simultane-

ously pronounced when the system enters an extreme state.

In empirical analysis, the tail regime is approximated by selecting exceedances above

a high quantile threshold. Let q denote the η-quantile of {R (t)}T , where η is typically

η uv t=1 164

chosen as a high level such as 0.90, 0.95, or 0.99. Define the exceedance set as

(cid:8) (cid:9)

T (η) = t ∈ {1, . . . , T} : R (t) > q . (3)

uv uv η 166

(u,v)

Then the empirical estimator of σ is given by

ij 167

^σ

i

(

j

u,v) =

|T

(η)|

∑ X

R

u,i

(

(

t

t

)

) · X

R

v,j

(

(

t

t

)

) . (4)

uv t∈T

uv

(η) uv uv

To obtain a scalar edge weight for network construction, we aggregate the TPDM entries as

w

^uv

=

m

∑ m ∑ m

^σ

i

(

j

u,v)

. (5)

i=1 j=1

中文:

本页进入 TPDM。作者把每个用户看成一个多维风险向量，例如发帖爆发强度、情绪极端程度、关键词爆发、回复扩张率、跨社区互动频率等。两个用户是否在危机状态下同步，不看普通平均相关，而看两个用户合并向量的“半径”很大时，各指标是否同时变大。公式 (1)-(5) 就是在定义极端状态、TPDM 条目和节点之间的标量边权。

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A larger value of w indicates stronger synchronized tail behavior between nodes u and v.

^uv 171

Based on w , we construct a weighted extremal dependence network. Let τ be the

^uv 172

edge inclusion threshold. The adjacency indicator is defined as

A = 1(w ≥ τ), (6)

uv ^uv 174

and the weighted adjacency matrix is

W~

uv

= A

uv

w

^uv

. (7)

The resulting graph captures synchronized extreme dependence rather than ordinary

interaction intensity, thereby providing a more suitable basis for identifying propagation

risk under rare but high-impact social events.

2.2. Interpretation of TPDM Edges

The edges in the TPDM-based network should be interpreted as extremal statistical

associations. A large value of w indicates that nodes u and v tend to exhibit synchronized

^uv 182

tail behavior under the selected threshold. It does not imply that node u causally transmits

information to node v, nor does it exclude the possibility that both nodes are simultaneously

driven by a common external shock.

This distinction is important for social information systems. During a public crisis, two

users or communities may become extreme at the same time because of direct interaction,

shared exposure to the same news event, platform-level recommendation, coordinated

amplification, or external media coverage. TPDM alone cannot distinguish among these

mechanisms. Therefore, TPDM edges in this paper are treated as candidate risk-coupling

channels that require additional temporal, semantic, or platform-level evidence before

being interpreted as causal propagation paths.

Accordingly, the proposed framework is positioned as a risk-screening and prior-

itization tool. It helps identify where extremal synchronization is concentrated in the

constructed network, but it does not provide causal identification by itself.

2.3. Parameter Selection and Sensitivity Diagnostics

The proposed framework involves three groups of parameters: the tail quantile level

η, the edge inclusion threshold τ, and the weights in the integrated risk score. These param-

eters influence the sparsity of the extremal dependence network, the resulting community

structure, and the final node ranking. Therefore, their selection should be made explicit.

For the tail quantile level η, a higher value focuses more strictly on extreme observa-

tions but reduces the number of exceedances used for estimating the TPDM. Conversely, a

lower value increases sample size but may include less extreme observations. In practice, η

can be examined over a reasonable range such as {0.90, 0.95, 0.975}, and the stability of the

resulting network can be evaluated by comparing edge density, dominant communities,

and top-ranked nodes.

The edge threshold τ controls the sparsification of the weighted extremal dependence

graph. In this paper, τ is selected by applying a percentile threshold to the empirical

distribution of w . A lower threshold produces a denser graph and may retain weak

^uv 209

extremal-dependence links, whereas a higher threshold removes weak links and may

generate peripheral singleton nodes. Sensitivity to τ can be assessed by reporting changes

in graph density, modularity, the number and size of detected communities, and the stability

of the minimum-cut set.

中文:

本页解释 TPDM 边和参数选择。边权大表示两个节点在极端状态下更容易一起出现高风险信号，但它不是因果边。作者用阈值 tau 把连续边权转成邻接矩阵，并把边权归一化。尾部分位数 eta 和边阈值 tau 是关键参数，需要敏感性分析。

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For the integrated risk score, the baseline setting uses equal weights,

α = β = γ = δ = 0.25,

which represents a neutral specification when there is no prior reason to privilege local

connectivity, brokerage, tail exposure, or community embedding. In applications where

domain knowledge is available, the weights can be adjusted to emphasize different mecha-

nisms. For example, a topology-oriented specification may assign larger weights to degree

and betweenness, whereas a tail-risk-oriented specification may assign larger weights to

tail exposure and community embedding. The robustness of node prioritization can be

evaluated by the overlap of top-ranked nodes and the Kendall rank correlation of R under

i 222

alternative weight vectors.

Accordingly, the following diagnostics are recommended when applying the frame-

work to empirical data:

1. graph-level diagnostics, including edge density, average weighted degree, and clus-

tering coefficient;

2. community-level diagnostics, including modularity, the number of communities, and

the size of dominant communities;

3. node-level diagnostics, including top-k node overlap and rank correlation of R ;

i 230

4. intervention-level diagnostics, including the stability of source–target communities

and minimum-cut edges.

These diagnostics help determine whether the main conclusions depend on a narrow

parameter choice or remain stable across reasonable parameter variations.

2.4. Network Centrality Indicators

To identify structurally critical nodes within the extremal dependence network, we em-

ploy degree centrality and betweenness centrality [11,16,17]. Degree centrality reflects local

connectivity and local amplification potential, whereas betweenness centrality captures the

extent to which a node serves as a bridge for cross-path transmission.

2.4.1. Degree Centrality

Degree centrality measures the number of direct neighbors a node has. In the present

setting, a high-degree node tends to interact with many other users and thus has a greater

potential to amplify propagation locally:

k

DC = i , (8)

i N − 1 244

where k is the degree of node i and N is the total number of nodes.

i 245

2.4.2. Betweenness Centrality

Betweenness centrality quantifies the extent to which a node lies on shortest paths

between other nodes:

BC = ∑ σ st (i) , (9)

i 249

σ

s̸=i, t̸=i st

s̸=t

where σ is the total number of shortest paths from node s to node t, and σ (i) is the

st st 250

number of those paths that pass through node i. The normalized betweenness centrality is

2BC

BN = i . (10)

i (N − 1)(N − 2) 252

中文:

本页给出参数诊断建议和两个基础中心性指标。degree centrality 衡量一个节点直接连了多少邻居，代表本地放大潜力；betweenness centrality 衡量一个节点是否经常位于其他节点最短路径上，代表跨路径桥接能力。

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Nodes with high BN can function as bridge nodes that transmit extreme influence across

i 253

otherwise weakly connected parts of the network.

2.4.3. Tail Exposure Centrality

While degree and betweenness describe topological importance, neither of them

directly reflects how strongly a node is synchronized with the rest of the network under

extreme conditions. To address this, we define the tail exposure centrality of node i as

1 ∑

TE

i

=

N − 1

W~

ij

. (11)

j̸=i

A high value of TE indicates that node i tends to co-fluctuate with many other nodes under

i 260

extreme states, suggesting a high level of extremal exposure.

2.5. Community Structure Detection

A community in a complex network is typically defined as a group of nodes with

dense internal connections and relatively sparse external connections [12,13]. In social

propagation networks, communities often correspond to interaction groups, topical clusters,

or latent opinion circles that share intensive internal exchanges.

From the perspective of extreme propagation, communities matter because abnormal

activity rarely remains isolated. If several users are tightly coupled, an extreme burst

originating from one node may induce correlated abnormal behavior within the same

community, while bridge edges or bridge nodes can transmit that shock across communities

and trigger system-wide cascades.

2.5.1. Role of Community Structure in Risk Localization

Community detection provides an intermediate granularity between node-level indi-

cators and global network summaries. It facilitates the identification of high-risk clusters in

which tail dependence is concentrated, enabling coordinated intervention. It also highlights

cross-community interfaces, where transmission pathways may be ambiguous and there-

fore require additional attention. In addition, community partitions allow analysts to move

beyond isolated user ranking and examine whether extreme propagation is concentrated in

specific groups.

2.5.2. Baseline Algorithm: Girvan–Newman

We employ the Girvan–Newman (GN) algorithm as the baseline community detection

method [12]. GN is a divisive, hierarchical algorithm based on edge betweenness. By

iteratively removing edges with the highest edge betweenness, it progressively reveals

densely connected communities. This property makes GN particularly suitable for the

present setting because it not only yields a community partition but also highlights bridging

edges that are natural candidates for subsequent monitoring, attenuation, or intervention

prioritization.

2.5.3. Partition Selection and Robustness Considerations

In practice, GN produces a sequence of partitions as edges are removed. A common

way to select the final partition is to maximize modularity [13,18]. However, modularity is

known to have a resolution limit in some settings. Therefore, when the network becomes

larger or when multi-scale structures are expected, it is advisable to complement GN with

alternative methods such as Louvain, Leiden, Infomap, or spectral clustering, and to check

the stability of the detected communities under reasonable parameter variations [19–22].

中文:

本页讨论社区发现。作者把极端依赖网络划分为社区，使用模块度 modularity 衡量社区划分好坏。社区不是普通朋友群，而是在极端状态下表现出更强内部耦合的一组节点。

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3. Extremal–Topological Risk Framework

Given the preliminary background, we now propose an integrated mathematical

framework for node-level risk quantification and community-level intervention. The

framework combines extremal dependence information with topological structure, which

therefore enables the multi-scale analysis of extreme propagation risk in complex social

information networks.

3.1. Integrated Risk Functionals

To quantify node-level propagation risk under extreme regimes, we construct an

integrated risk functional that combines four network mechanisms: local connectivity,

global brokerage, tail exposure, and community embedding strength. This formulation

explicitly relates extremal dependence structure with network topology, facilitating a

multi-scale characterization of propagation risk.

Let D~C

i

, B~C

i

, and T~E

i

denote the normalized degree centrality, normalized between-

ness centrality, and normalized tail exposure centrality of node i, respectively. Min–max

normalization is adopted:

X − min X

X~i =

max X

i

− min

j

X

j

+ ε

, X

i

∈ {DC

i

, B

i

N, TE

i

, CE

i

}, (12)

j j j j

where ε > 0 is a small constant introduced to avoid division by zero.

To incorporate mesoscopic community effects into node-level risk estimation, we

define the community embedding strength of node i as





|C(i)

| − 1

∑ W~

ij

, |C(i)| > 1,

CE = j∈C(i) (13)

i j̸=i 314

0,

|C(i)| = 1.

where C(i) denotes the community to which node i belongs. This quantity measures how

strongly node i is coupled to other members within its own community under extreme

conditions. For singleton communities, CE is set to zero because no intra-community

i 317

neighbor exists.

Let C~E

i

denote the normalized value of CE

i

. The integrated risk score is then defined

as

R

i

= αD~C

i

+ βB~C

i

+ γT~E

i

+ δC~E

i

, (14)

subject to

α + β + γ + δ = 1, α, β, γ, δ ≥ 0. (15)

In this formulation, D~C

i

captures the local amplification potential of node i, B~C

i 324

reflects its role as a cross-path bridge, T~E

i

measures its extremal synchronization with the

overall network, and C~E

i

characterizes its embeddedness in a tightly coupled high-risk

community. Compared with a model that only combines degree and betweenness, the

proposed formulation is better aligned with the objective of identifying propagation risk

under extreme regimes.

At the community level, we define the internal extremal coupling strength of commu-

nity C as

k 331

 1 ∑



|E |

W~

uv

, |E

k

| > 0,

Q k = k (u,v)∈Ek (16) 332

0,

|E | = 0,

k

中文:

本页提出综合风险函数。每个节点风险由四部分组成：度中心性、介数中心性、尾部暴露中心性、社区嵌入强度。先把不同指标归一化，再用权重 alpha、beta、gamma、delta 加权求和。社区层面还定义内部极端耦合强度 Q_k。

<a id="S009"></a>

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Source: p.9 S009

Original:

where E is the set of intra-community edges in community C . The second case is intro-

k k 333

duced for singleton or edgeless communities, whose internal extremal coupling is set to

zero by definition. This quantity will be used in the next subsection to identify extreme

source communities.

In implementation, the weight parameters may be selected based on sensitivity analy-

sis, expert knowledge, or cross-validated optimization. In the empirical study below, we

adopt

α = 0.25, β = 0.25, γ = 0.25, δ = 0.25

as a baseline setting, and further examine the stability of the rankings under alternative

parameter combinations.

3.2. Regularity Properties of the Proposed Risk Functional

We now report basic regularity properties of the proposed risk functional and the

associated extremal exposure measures. These properties support the numerical stability

and interpretability of the scoring rule, but they should not be interpreted as empirical

validation of real-world propagation mechanisms.

Theorem 1. Consider R as defined in Eq. (14), where all normalized features lie in [0, 1], and the

i 348

weights satisfy

α + β + γ + δ = 1, α, β, γ, δ ≥ 0.

Then the following properties hold:

1. Boundedness:

0 ≤ R ≤ 1.

i 353

2. Componentwise monotonicity: if for two nodes i and j,

D~C

i

≥ D~C

j

, B~C

i

≥ B~C

j

, T~E

i

≥ T~E

j

, C~E

i

≥ C~E

j

,

then

R ≥ R .

i j 357

Moreover, if at least one of the above inequalities is strict and the corresponding weight is

strictly positive, then R > R .

i j 359

3. Lipschitz continuity: let

x

i

= (D~C

i

, B~C

i

, T~E

i

, C~E

i

)⊤ , w = (α, β, γ, δ)⊤ .

Then for any two nodes i and j,

|R

i

− R

j

| ≤ ∥w∥

∥x

i

− x

j

∥∞ = ∥x

i

− x

j

∥∞.

4. Ranking stability under perturbation: suppose the normalized feature vector is perturbed

to

x

i

′ = x

i

+ ε

i

, ∥ε

i

∥∞ ≤ η.

Let R′ = w⊤x′. If for two nodes i and j,

i i 367

R − R > 2η,

i j 368

then their ordering is preserved after perturbation, namely

R ′ > R ′ .

i j 370

中文:

本页给出风险函数的规则性质。作者想说明这个分数不是随便拼出来的：它有界，指标越高分数不会反而下降，对小扰动有 Lipschitz 稳定性，排名在扰动不超过间隔一半时保持稳定。对没有数学基础的人来说，这一页是在证明“这个打分公式至少不会明显乱来”。

<a id="S010"></a>

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Proof. Since each normalized feature lies in [0, 1] and the weights are nonnegative with

unit sum, R is a convex combination of bounded quantities. Therefore,

i 372

0 ≤ R ≤ 1,

i 373

which proves boundedness.

For monotonicity, observe that

R

i

− R

j

= α(D~C

i

− D~C

j

) + β(B~C

i

− B~C

j

) + γ(T~E

i

− T~E

j

) + δ(C~E

i

− C~E

j

).

As long as each term on the right-hand side is nonnegative, we have R − R ≥ 0. If at least

i j 377

one difference is strictly positive and the corresponding weight is strictly positive, then

R − R > 0.

i j 379

For Lipschitz continuity, write

R − R = w ⊤(x − x ).

i j i j 381

By Hölder’s inequality,

|R

i

− R

j

| ≤ ∥w∥

∥x

i

− x

j

∥∞.

Since α + β + γ + δ = 1, then ∥w∥ = 1 so that

1 384

|R

i

− R

j

| ≤ ∥x

i

− x

j

∥∞.

Under perturbation, we see that

R ′ − R ′ = (R − R ) + w ⊤(ε − ε ).

i j i j i j 387

Applying Hölder’s inequality again gives

|w ⊤(ε

i

− ε

j

)| ≤ ∥w∥

∥ε

i

− ε

j

∥∞ ≤ 2η.

Hence,

R ′ − R ′ ≥ (R − R ) − 2η,

i j i j 391

and we conclude that if R − R > 2η, then R′ − R′ > 0. This proves the stability of

i j i j 392

ranking.

Theorem 1 shows that the proposed risk functional is a well-defined convex aggrega-

tion rule with boundedness, monotonicity, and ranking robustness. Its role is to justify the

stability of the prioritization index under normalized inputs, not to prove that the inferred

network represents causal diffusion in empirical social systems.

We now also give the monotonicity of extremal exposure measures.

Proposition 1. Assume W~

ij

≥ 0 for all i, j. Then:

1. The tail exposure centrality

1 ∑

TE

i

=

N − 1

W~

ij 401

j̸=i

and the community embedding strength





|C(i)

| − 1

∑ W~

ij

, |C(i)| > 1,

CE = j∈C(i)

i j̸=i 403

0,

|C(i)| = 1

中文:

本页继续证明和解释尾部暴露 TE 与社区嵌入 CE。TE 关注节点与全网其他节点的极端依赖总强度，CE 关注节点与同社区成员的极端耦合。两者都来自 TPDM 边权，所以比只看普通拓扑更贴近极端传播风险。

<a id="S011"></a>

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Source: p.11 S011

Original:

are nonnegative.

2. Both TE and CE are monotonically increasing with respect to increases in the extremal

i i 405

dependence weights.

Proof. Nonnegativity follows directly from the assumption W~

ij

≥ 0 and the linearity of

summation.

For monotonicity, increasing W~

ij

by ∆ > 0 gives

∆

TE (cid:55)→ TE + .

i i N − 1 410

If, in addition, j ∈ C(i) and |C(i)| > 1, then

∆

CE (cid:55)→ CE + .

i i |C(i)| − 1 412

For singleton communities, CE remains zero by definition.

i 413

Hence, both TE and CE are monotone with respect to increases in the extremal

i i 414

dependence weights.

Proposition 1 shows that the exposure measures respond monotonically to changes

in the estimated extremal dependence weights. This provides a consistency check for the

scoring construction, while the empirical meaning of these weights still depends on the

validity of the data, threshold selection, and modelling assumptions.

3.3. Community-Level Propagation Blocking via Minimum Cut

Community detection identifies groups of nodes with dense internal dependence and

relatively sparse external connections. However, identifying high-risk communities alone is

insufficient if the ultimate goal is to design intervention strategies. To move from diagnosis

to control, we aggregate the original node-level network into a community graph and

formulate a minimum-cut optimization problem.

Let the detected communities be denoted by {C , C , . . . , C }. We define the commu-

1 2 K 426

nity graph as

G (c) = (V (c) , E (c) , Ω),

where each super-node in V(c) corresponds to a community. For two distinct communities

C and C , the inter-community coupling strength is defined as

a b 430

Ω

ab

=

|E

|

∑ ∑ W~

uv

1 ( (u, v) ∈ E

ab

) , (17)

ab u∈Ca v∈Cb

where E denotes the set of edges connecting communities C and C .

ab a b 432

We then define the set of source communities S and target communities T . Source

communities represent extreme or highly risky communities, while target communities

represent mainstream or highly exposed communities. A practical selection rule is

S = {C : Q ranks among the top k and R¯ ranks among the top k }, (18)

k k s k s 436

where

R¯ = 1 ∑ R (19)

k |C | i 438

k i∈Ck

is the average node risk within community C . The target set T may be defined as the set of

k 439

the largest normal communities or the communities with relatively low internal extremal

intensity but high external exposure.

中文:

本页进入社区级阻断问题。作者先把节点社区压缩成社区图，社区之间的边权 Omega_ab 表示两个社区之间的极端耦合强度。然后根据社区内部强度 Q_k 和平均节点风险 Rbar_k 选择 source communities 与 target communities。

<a id="S012"></a>

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Original:

To identify candidate intervention interfaces between S and T , we solve the following

community-level minimum-cut problem. Let z ∈ {0, 1} indicate whether the inter-

ab 443

community interface (a, b) is selected for attenuation in the constructed community graph.

The optimization model is

∑

min c z , (20)

ab ab 446

z

(a,b)∈E(c)

subject to

all paths from S to T are separated in the constructed community graph after selecting interfaces with z = 1,

ab

(21)

z ∈ {0, 1}, (a, b) ∈ E (c) . (22)

ab 450

The multi-source multi-target intervention problem can be equivalently transformed

into a standard minimum s–t cut problem by introducing a super-source node s connected

to each community in S and a super-sink node t connected from each community in T .

The resulting augmented graph preserves the feasibility and optimality structure of the

original community-level separation problem.

Theorem 2. Let G(c) be the community-level extremal dependence graph with edge costs c > 0.

ab 456

Then the intervention-interface selection problem in Eq. (20) is equivalent to a minimum s–t cut

problem on the augmented community graph. The optimal solution identifies the minimum-cost set

of inter-community interfaces whose attenuation separates the selected source communities from the

selected target communities in the constructed graph.

The proof of Theorem 2 follows from standard minimum-cut theory. The constraint

that all source–target paths are separated in the constructed community graph is equivalent

to separating S and T after selecting the corresponding community-level interfaces. By

introducing a super-source node s connected to all communities in S and a super-sink node

t connected from all communities in T , the problem reduces to a standard minimum s–t

cut formulation. The max-flow min-cut theorem then ensures that an optimal cut can be

obtained by a maximum-flow computation.

The term “attenuation” is used deliberately. In real social information systems, a

community-level cut does not mean that social ties are physically removed or that infor-

mation diffusion is guaranteed to stop. Instead, the cut set should be interpreted as a

candidate set of interfaces where intervention resources may be prioritized. Depending on

the platform context, such intervention may correspond to enhanced monitoring, reduced

algorithmic amplification, targeted fact-checking, delayed recommendation, or human

moderation.

Proposition 2. The community-level extremal minimum-cut problem is dual to a maximum-flow

problem on the augmented community graph. Consequently, the minimum cut identifies bottleneck

inter-community interfaces in the constructed graph.

By the max-flow min-cut theorem, the minimum cut is dual to a maximum-flow

problem. In the present framework, this duality is used as a graph-theoretic diagnostic

rather than as a direct model of real information transmission capacity.

In the absence of real operational cost data, we use a proxy cost to reflect intervention

priority:

c = , ε > 0. (23)

ab Ω + ε 483

ab

中文:

本页把干预接口选择写成 minimum-cut 优化问题。变量 z_ab 表示是否选择某条社区间接口进行削弱，目标是在切断 source 到 target 所有路径的前提下让成本最小。作者强调 attenuation 是削弱或优先干预，不是物理删除社交关系，也不保证信息完全停止。

<a id="S013"></a>

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Source: p.13 S013

Original:

This proxy assigns lower cost to stronger extremal coupling, meaning that highly coupled

community interfaces are prioritized for attenuation in the synthetic experiment. The

resulting minimum cut should therefore be understood as a model-based prioritization

result rather than as an operational guarantee of propagation blockage.

4. Results

4.1. Controlled Synthetic Experiment and Validation Scope

To evaluate the internal consistency of the proposed framework, we constructed a con-

trolled synthetic online discussion network with 100 nodes. Each node represents an active

user, and edges represent interaction relationships such as replying, mentioning, reposting,

or repeated co-participation within the same discussion thread. The observation period

contains 180 time steps, and each node is described by a five-dimensional propagation-risk

vector, including posting burst intensity, sentiment extremeness, keyword burst level, reply

expansion rate, and cross-community interaction frequency.

For the quantitative edge-recovery benchmark, we further inject 15 predefined cross-

community high-risk links between the source-like and target-like communities. These

injected links are exposed to synchronized heavy-tailed shocks during selected crisis win-

dows, while non-risk pairs receive only background fluctuations or non-synchronous noise.

The injected links form the gold-standard set E used only for controlled evaluation.

gold 501

They should not be interpreted as real social-media ground-truth propagation links.

The use of synthetic data has two purposes. First, it allows the extremal dependence

structure and the community-level source–target relation to be examined under controlled

conditions. Second, it avoids privacy, sampling, and platform-access constraints that often

arise in real social media data. However, this design also implies that the present experiment

should not be interpreted as a direct real-world validation. Instead, it provides a proof-

of-concept test of whether the proposed TPDM-based network construction, integrated

risk ranking, community detection, and minimum-cut intervention model can operate as a

coherent analytical pipeline.

Based on the simulated time-indexed risk vectors, we estimated pairwise TPDM

values and constructed the weighted extremal dependence network according to Eqs. (4)–

(7). Under the baseline tail quantile setting of η = 0.95 and the edge percentile threshold

of 85%, the estimated edge inclusion threshold was τ = 0.084926. After thresholding,

the resulting extremal dependence graph contained 743 edges, indicating that extreme

synchronization was concentrated in a relatively dense but still structured subset of node

pairs.

4.2. Risk Quantification and Visualization

Using the TPDM-based dependence weights W~ = (W~

uv

), we characterized how

extreme propagation states synchronize across users under the tail regime. The resulting

TPDM heatmap (Figure 1) shows a clear block structure, indicating that synchronized

extreme behavior is concentrated in a limited number of densely coupled regions rather

than being uniformly distributed across the network. This suggests that, under the synthetic

setting, extremal synchronization is concentrated in a small number of candidate risk-

coupling channels.

Table 1 presents the top-ranked nodes based on the integrated risk score R . The top

i 526

five high-risk nodes are User_23, User_58, User_62, User_60, and User_53, and all of them

belong to Community 0. This result indicates that the most structurally critical nodes

are concentrated in the dominant high-risk community rather than scattered across the

network.

中文:

本页开始结果部分。实验是 100 个节点的合成社交讨论网络，有潜在社区结构和模拟的极端活动。作者按 eta=0.95 和边权阈值构造 TPDM 网络，目的是展示方法流程，而不是声称真实平台有效。

<a id="S014"></a>

Page 14

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Original:

Table 1. Top 5 high-risk nodes under the integrated risk model.

Node

Rank

ID

Comm. D~C

i

B~C

i

T~E

i

C~E

i

R

i

1 User_23 0 0.9375 1.0000 0.9394 0.8780 0.9387

2 User_58 0 1.0000 0.3572 1.0000 1.0000 0.8393

3 User_62 0 0.9375 0.2555 0.9366 0.9366 0.7665

4 User_60 0 0.8958 0.1839 0.8950 0.8746 0.7123

5 User_53 0 0.8542 0.0776 0.8479 0.8479 0.6569

0 20 40 60 80 99

Node index (ordered by community)

)ytinummoc

yb

deredro(

xedni

edoN

TPDM Heatmap Ordered by Detected Community

C0 source (55) C1 target (35) iso(1la0t)es

0.0875

0.0850

0.0800

0.0750

0.0700

55 90

thgiew

MDPT

detamitsE

Figure 1. TPDM heatmap of the synthetic social discussion network. Darker colors indicate stronger

tail dependence. The block structure suggests that extreme synchronization is concentrated in a few

dominant regions of the network.

4.3. Baseline Comparison with Average-Dependence Networks

To address whether the TPDM-based representation provides additional information

beyond average-dependence measures, we compare the proposed method with Pearson-

correlation and Spearman-correlation edge rankings under the same synthetic setting. The

benchmark contains N = 100 nodes, T = 180 time steps, five propagation-risk dimensions,

and 15 injected cross-community high-risk links. For each method, the top-k predicted

links are compared with the injected gold-standard links, where k = 20. The experiment is

repeated over 100 Monte Carlo replications with different random seeds.

The evaluation metrics are defined as

|E^risk ∩ E

gold

| |E^risk ∩ E

gold

|

Precision = , Recall = , (24)

|E^risk | |E

gold

| 540

and

2 · Precision · Recall

F1 = . (25)

Precision + Recall 542

Here, E^risk denotes the set of top-ranked predicted high-risk links and E

gold

denotes the set

of injected gold-standard links.

中文:

本页展示风险排序、TPDM 热图和与平均依赖网络的比较指标。结果显示 TPDM 网络能更好恢复预设的风险耦合结构。Table 1 是高风险节点，Figure 1 是按社区排序的 TPDM 热图。

<a id="S015"></a>

Page 15

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Original:

Table 2. Baseline comparison on the controlled synthetic benchmark. Values are reported as mean ±

standard deviation over 100 Monte Carlo replications.

Method Hits Precision Recall F1-score

0.7480 ± 0.9973 ± 0.8549 ±

TPDM-based network 14.96 ± 0.20

0.0098 0.0131 0.0113

0.6840 ± 0.9120 ± 0.7817 ±

Pearson correlation 13.68 ± 1.14

0.0568 0.0757 0.0649

0.0860 ± 0.1147 ± 0.0983 ±

Spearman correlation 1.72 ± 1.22

0.0608 0.0810 0.0694

As shown in Table 2, the TPDM-based network achieves the highest recovery perfor-

mance in the controlled benchmark. Compared with Pearson correlation, TPDM improves

the mean F1-score from 0.7817 to 0.8549. The improvement is not presented as universal em-

pirical superiority, but as evidence that the injected high-risk links in this synthetic design

are more consistently recovered when edge ranking is based on tail-regime co-exceedance.

Spearman correlation performs poorly because the injected crisis shocks are sparse in time

and are partially masked by non-crisis fluctuations, making full-sample rank association

less informative in this setting.

4.4. Community Structure and Source–Target Identification

Using the Girvan–Newman algorithm, we examined the mesoscopic structure of the

extremal dependence network. The results revealed two dominant communities and a

set of peripheral singleton communities. Specifically, Community 0 contains 55 nodes

and Community 1 contains 35 nodes, whereas the remaining 10 communities consist of

only one node each. This indicates that, under the current thresholding strategy, the

network structure is dominated by two major coupled groups together with several weakly

connected peripheral users.

At the community level, Community 0 exhibits the highest internal extremal coupling

strength (Q = 0.086389) and the highest average node risk (R¯ = 0.323057), and is therefore

0 0 562

identified as the source community. Community 1 has a comparable internal coupling

strength (Q = 0.086352) but a lower average risk (R¯ = 0.209541), and is identified as the

1 1 564

target or mainstream exposure community. The remaining singleton communities have

zero intra-community coupling after thresholding and are therefore treated as peripheral

isolated nodes rather than substantive propagation groups.

Table 3. Dominant community-level statistics for source–target selection.

External

Community Size Q R¯ Role

k k Exposure

Source /

C 55 0.086389 0.323057 0.085378

0 Extreme

Target /

C 35 0.086352 0.209541 0.085378

1 Mainstream

To avoid over-interpreting threshold-induced singleton partitions, the 10 remaining

one-node communities are not listed individually in Table 3. They are treated as isolated

peripheral users with weak extremal connectivity under the current edge sparsification

setting.

中文:

本页继续基线比较，并进入社区结构。Table 2 报告 precision、recall、F1 等指标，TPDM 版本优于 average-dependence baseline。随后作者识别出两个主导社区和若干孤立节点。

<a id="S016"></a>

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Original:

4.5. Minimum-Cut Intervention Strategy

After constructing the community graph, we solved the minimum-cut problem to

identify the least-cost candidate intervention interface between the source community and

the target community in the constructed graph. The optimization result is structurally

simple and interpretable: the selected cut consists of the single community-level interface

between Community 0 and Community 1. The corresponding inter-community coupling

strength is Ω = 0.085378, and the proxy cut cost is 11.712609.

01 578

This result indicates that the main cross-community extremal coupling is concentrated

between the two dominant communities. In other words, the extreme opinion propagation

pattern in this experiment is not represented by many competing cross-community routes,

but rather by one primary inter-community coupling interface. Such a structure provides

a clear candidate interface for targeted monitoring or attenuation. However, this result

should be interpreted on the constructed community graph and does not imply that real-

world diffusion can be fully blocked by acting on a single channel.

Table 4. Community-level minimum-cut result.

Source

Target Community Ω Cut Cost

Community ab

C C 0.085378 11.712609

0 1

Community 0 Community 1

Highlighted minimum-cut channel

Top bridge-risk nodes C0 to C1

(User 23, 58, 60, 62, 53)

Community 0: source Community 1: target Singleton node Minimum-cut channel

Node size is proportional to the integrated risk score; labeled nodes are the top-ranked bridge-risk nodes.

Figure 2. Extremal dependence network of the synthetic social discussion system. Node size is

proportional to the integrated risk score, and node color indicates the detected community label. The

figure shows two dominant coupled groups together with several peripheral singleton nodes.

4.6. Sensitivity and Robustness Considerations

The baseline results are obtained under η = 0.95 and an edge percentile threshold

of 85%. Since both the TPDM estimator and the sparsified extremal dependence graph

depend on these choices, the results should be interpreted together with the thresholding

mechanism. A lower edge threshold would retain more weak extremal-dependence links

中文:

本页给出 source-target 社区选择和 minimum-cut 结果。Table 3 说明社区 0 和社区 1 是主要风险相关社区；Table 4 的 minimum cut 结果指出 C0-C1 是主要候选干预接口。

<a id="S017"></a>

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Original:

and produce a denser graph, whereas a higher threshold would remove weak links and

may generate additional peripheral singleton communities.

In the present experiment, the main qualitative finding is not the exact number of

singleton communities, which may vary with the sparsification threshold. Rather, the

substantive pattern is that the extremal dependence structure is concentrated around

two dominant coupled communities, with the highest-risk nodes located in the source

community and the principal inter-community channel connecting the source community

to the target community. Therefore, the interpretation should focus on the dominant

source–target relation instead of over-interpreting threshold-induced peripheral partitions.

The equal-weight specification in Eq. (14) is used as a neutral baseline. In practical

applications, alternative weight vectors can be examined to test whether the top-ranked

nodes remain concentrated in the same high-risk community. Similarly, Girvan–Newman

is adopted as an interpretable baseline community detection method, but larger or denser

networks should be checked using scalable alternatives such as Louvain or Leiden. These

robustness checks are especially important when the framework is transferred from con-

trolled synthetic data to real social media datasets.

5. Discussion

5.1. Main Findings and Methodological Implications

The controlled experiment shows that the proposed framework provides a coherent

structural description of extreme opinion propagation under a synthetic setting. First,

the TPDM heatmap reveals a block-like structure, indicating that synchronized extreme

behavior is concentrated in a limited number of densely coupled regions rather than being

uniformly distributed across the network. This supports the methodological claim that tail

dependence can provide information that is different from average co-movement.

The baseline comparison further supports this interpretation. In the controlled edge-

recovery benchmark, the TPDM-based method obtains the highest mean F1-score among

the three tested dependence measures. This result suggests that, when high-risk links are

generated through sparse synchronized crisis shocks, tail-regime co-exceedance can be

more informative than full-sample average dependence. However, the comparison remains

limited to the synthetic design and should not be interpreted as evidence of universal

superiority in empirical social media networks.

Second, the node-level ranking result indicates that the highest-risk users are concen-

trated in Community 0. In particular, User_23, User_58, User_62, User_60, and User_53

rank at the top of the integrated risk list. This pattern suggests that the dominant source of

structural risk in the synthetic network is localized in a compact portion of the graph. The

result also illustrates the value of combining local connectivity, brokerage, tail exposure,

and community embedding strength in a single risk score.

Third, the community detection result should be interpreted cautiously. The network

is dominated by two major communities and a set of peripheral singleton nodes. This

does not imply that real social information networks generally have only two substantive

propagation groups. Rather, it reflects the structure induced by the current synthetic setting

and thresholding strategy. Therefore, the singleton communities are better understood as

weakly connected peripheral users under the present sparsification threshold rather than

as meaningful propagation modules.

Finally, the minimum-cut result demonstrates how the proposed framework can move

from diagnosis to intervention. In the present experiment, the optimal cut consists of one

inter-community channel between C and C , indicating that the source–target relation is

0 1 637

highly concentrated. This finding should not be overgeneralized to more complex empirical

networks, where multiple communities and multiple cross-community routes may exist.

中文:

本页讨论敏感性和主要发现。作者说 eta、tau 和权重选择会影响结果，因此应检查 top-k 节点重叠、Kendall 排名相关、社区数量和 minimum-cut 边稳定性。主要贡献是把极端同步结构转化为可分析的网络对象。

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The main contribution of the minimum-cut component is therefore methodological: it

provides a mathematically explicit way to translate community-level extremal coupling

into a least-cost intervention-interface selection problem.

The practical implications of the framework should also be stated with caution. The

results suggest that TPDM-based networks may be useful for identifying candidate bridge

nodes and candidate inter-community channels for further monitoring. However, the

present study does not directly validate operational early warning, content moderation, or

platform-level intervention on real social media data. Such applications require additional

empirical testing, ethical review, and platform-specific implementation constraints.

5.2. Failure Cases and Uncertainty

Several failure cases should be considered when interpreting the proposed framework.

First, a high TPDM value may be generated by a common external shock rather than by di-

rect or indirect user-to-user influence. For example, a major news event, coordinated media

coverage, or a platform-wide recommendation change may cause multiple communities

to become extreme simultaneously. In such cases, TPDM correctly detects extremal co-

movement, but the corresponding edge should not be interpreted as a causal propagation

path.

Second, the minimum-cut result may be unstable when the community graph contains

several routes with similar coupling strengths or similar proxy costs. In the present synthetic

experiment, the cut set contains one dominant community-level interface, but this simplicity

may not persist in larger or more heterogeneous networks. In empirical settings, multiple

near-optimal cut sets should be reported to avoid over-interpreting a single intervention

solution.

Third, social media users may adapt after intervention. Even if one candidate channel

is attenuated, users may switch topics, migrate to other communities, use alternative

platforms, or rely on indirect reposting chains. Therefore, the cut set should be interpreted

as a prioritization tool for intervention planning rather than as a deterministic blocking

mechanism.

Fourth, TPDM estimation is sensitive to the number of exceedance observations. If

the tail threshold is too high, the estimator may become unstable due to insufficient tail

samples. If the threshold is too low, the method may capture ordinary fluctuations rather

than extreme behavior. This uncertainty should be addressed through threshold sensitivity

analysis, repeated simulation, confidence intervals, or bootstrap-based stability diagnostics

in future empirical studies.

6. Conclusions

This paper proposed a community-structured extremal dependence framework for

analyzing extreme opinion propagation risk in social information networks. The frame-

work combines TPDM-based network construction, composite node-level risk scoring,

community detection, and a community-level minimum-cut formulation.

The controlled synthetic experiment shows that the proposed pipeline can produce an

interpretable structural representation under known modelling conditions. In the current

setting, the extremal dependence network is dominated by two major coupled communities

and several peripheral singleton nodes. The highest-ranked nodes are concentrated in

the source community, and the community-level cut identifies one principal source–target

interface in the constructed graph.

The quantitative edge-recovery benchmark further compares the TPDM-based net-

work with Pearson and Spearman correlation baselines. Under the designed crisis-shock

mechanism, the TPDM-based method achieves the highest mean F1-score, suggesting

中文:

本页讨论失败场景、局限和结论。风险包括：真实数据噪声、非平稳、平台机制变化、机器人行为、多通道传播、边被误读为因果、minimum cut 被误读成完全阻断。结论强调该方法是候选风险耦合结构识别框架，未来需要真实数据和因果时间信息验证。

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that tail-regime co-exceedance is useful for recovering the injected high-risk links in this

controlled synthetic setting.

These findings support the methodological feasibility of combining extremal depen-

dence estimation with network topology and community-level optimization. However,

they do not establish real-world causal propagation or operational intervention effective-

ness. TPDM edges should be interpreted as candidate extremal risk associations, and the

minimum-cut result should be interpreted as a candidate intervention interface rather than

as proof that information diffusion can be fully blocked.

Future work should validate the framework on public or platform-authorized social

media datasets, incorporate topology-only and temporal-causal baselines, integrate seman-

tic information, and examine robustness under external shocks, multi-channel diffusion,

and adaptive user behavior.

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本页是参考文献，包含社会动力学、社交网络意见动力学、谣言与虚假新闻传播、复杂网络、极值图模型和社区发现等基础文献。

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中文:

本页继续参考文献，主要包含社区发现算法、Leiden/Louvain/Infomap、谱聚类等网络社区检测相关文献。

6. 这篇论文和本仓库学习方向的关系

这篇论文连接了三个方向：极值理论负责“只看极端时刻”，复杂网络负责“把用户关系变成图”，风险传播负责“找高风险节点和社区接口”。如果以后课程作业要求做极端事件、社交网络或复杂系统建模，可以把它拆成：定义风险指标、构造极端依赖矩阵、建图、计算中心性、做社区发现、解释干预意义。

7. 可以追问老师/同学的问题

• TPDM 和 tail dependence coefficient 有什么关系？这里的 TPDM 是否满足标准极值理论定义？
• 合成实验中真实风险边是如何生成的？precision/recall 的 ground truth 是否合理？
• minimum cut 的成本设为 1/(Omega+epsilon) 是否会导致强耦合边更容易被选中，这在真实干预中是否符合成本逻辑？
• 如果有时间顺序，如何区分极端同步和因果传播？
• eta 和 tau 如何系统调参，而不是只凭经验选择？

8. 给 AI 的高质量提示词

请把这篇论文当作“极值理论 + 复杂网络 + 社交舆论风险传播”的方法论文来读。先解释 TPDM 为什么比 Pearson correlation 更适合极端传播，再逐条解释公式 (1)-(25)，最后给出一个可以用 Python 复现的最小实验流程。要求面向跨专业学生，先直觉解释，再专业表达，再公式解释。