Threshold cascade dynamics on signed random networks

2025-05-06 0 0 554.6KB 9 页 10玖币

侵权投诉

Kyu-Min Leea,b, Sungmin Leea, Byungjoon Minc, K.-I. Goha

aDepartment of Physics, Korea University, Seoul 02841, Korea

bCollege of Business, Korea Advanced Institute of Science and Technology, Seoul 02455, Korea

cDepartment of Physics, Chungbuk National University, Cheongju 28644, Korea

Abstract

Relationships between individuals in a social network, genes in biological systems, and spins in magnetic systems often

reﬂect a mixture of positive (friendly) and negative (antagonistic) interactions. However, most studies of complex

networks have focused on networks consisting of solely positive interactions. Here, we study threshold cascades on

signed networks composed of both positive and negative connections, focusing on when a pair of nodes connected by a

negative link can only be activated exclusively to each other. We found that the negative interactions not only suppress

global cascades, but also induce the heterogeneity in activation patterns manifesting from single-node to network levels.

Our results suggest that negative interactions may be an important source of the variability in cascading dynamics.

Keywords: Signed networks, Threshold cascade, Negative links, Heterogeneous activation patterns

1. Introduction

Modeling how the cascades of activations occur in

threshold-based dynamics is fundamental for understand-

ing collective behaviours in social and biological complex

systems [1, 2, 3, 4]. In order to model the cascading phe-

nomena triggered by a tiny perturbation, a threshold cas-

cade model was proposed [2, 3]. This model was originally

motivated by the behavioral and emotional contagions in

a society where individuals are encouraged to follow what

their connected neighbors are doing. In addition, thresh-

old cascades driven by integrate-and-ﬁre mechanisms are

associated with the avalanches of neural activations [5, 6],

the spread of economic crisis [7], and cascading failures in

infrastructure networks [8, 9, 10]. The key mechanism in

this model is that nodes are activated when the fraction of

activated neighbors exceeds their threshold assigned a pri-

ori. In this model, cascades with an extensive size, called

global cascades, can occur from an extremely small frac-

tion of seeds because the cascades of activations propagate

along connected neighbors [3, 11].

In threshold cascade models on networks, links act as

channels for cascade propagation, such that the inﬂu-

ence or stimulus arriving from each neighbor contributes

positively to reaching the threshold [3]. Although tra-

ditional cascade modeling, which consists exclusively of

positive links [3, 11, 12, 13, 14, 15], renders the model

simple and tractable, it overlooks the negative interac-

tions in the cascade dynamics. Adversarial interactions

are common and essential elements of many networked

systems [16, 17, 18, 19, 20]. “Dislike” relationships in

Email addresses: bmin@cbnu.ac.kr (Byungjoon Min),

kgoh@korea.ac.kr (K.-I. Goh)

social networks [17, 18, 20, 21, 22], inhibitory signals

in genetic regulation [16, 23, 24], synaptic inhibition in

neural networks [19], antagonistic competitions between

nations [25, 26], and antiferromagnetic bonds in mag-

netic systems [27] are typical examples of adversarial re-

lationships, to name a few. Not only negative links are

widespread in real-world systems, but also they play a

qualitatively diﬀerent role in dynamical processes than

positive links [16, 19]. Networks with both types of inter-

actions can be better modeled as “signed networks” where

links are either positive or negative [18, 28, 29, 30, 33]. The

concept of signed networks has long been proposed in psy-

chology and sociology, through social balance theory [28]

and structural balance theory [29, 32, 31]. In addition,

from the perspective of statistical physics, coexistence of

positive and negative interactions has important implica-

tions as a source of geometric frustration and dynamic

heterogeneity [27, 34]. As such, the studies on signed net-

works have received due attention from statistical physics

and network science communities [30, 35, 36]. However,

studies on the impact of negative interactions on thresh-

old cascade dynamics are still lacking.

In this work, we study the dynamics of a threshold cas-

cade model on signed random networks or “signed” cas-

cade, to be short. In our “signed” cascade model, nodes’

activation is completely blocked if there exist active adver-

sarial neighbors. That is, no pair of nodes connected by a

negative link can be activated at the same time. It models,

in an idealized way, the following real-world scenarios: In

the case of “distrust” or “dislike” relationships denoted by

negative links in a signed social network, someone would

never agree and follow with their negative-linked person’s

opinion or behavior, regardless of what their friends are

Preprint submitted to Elsevier February 7, 2023

arXiv:2210.02011v2 [physics.soc-ph] 6 Feb 2023

positive

negative

inactive

active

seed

Figure 1: Illustration of threshold cascades on a signed network. If

the fraction of active neighbors of node iconnected by positive links

is larger than threshold θ= 0.4 and there are no active neighbors

connected by negative links, node ibecomes active. Note that there

can be various scenarios of cascades activations depending on the

sequence of activations.

doing. In such cases, a pair of nodes connected by a neg-

ative link cannot become active at the same time. Theo-

retically speaking, our model implements the cascade dy-

namics where negative links co-operate with positive links

in functionally-multiplicative manner rather than additive

manner.

There are some related previous studies in statistical

physics literature on signed networks such as random

threshold networks [4], percolation of antagonistic multi-

plex networks [37], opinion models on evolving signed net-

works [38], epidemic spreading on signed networks [39, 40],

and threshold model with anticonformity [41, 42]. Infor-

mation diﬀusion and linear threshold models in signed net-

works have also received much attention and how the in-

formation diﬀusion dynamics in signed networks depends

on the diﬀusion path and structural balance was studied

[43, 44, 46, 47]. However, in contrast to our model most

previous studies have considered the eﬀect of negative links

additively [4, 41, 43, 40] or focused on structural properties

rather than dynamical consequences [37]. In this study, we

implemented cascading dynamics in a way that maximizes

the multiplicative coupling eﬀect of negative links. We

conﬁrmed that the negative links can signiﬁcantly reduce

the size of the global cascades. We also found that negative

interactions can produce the heterogeneity in the activa-

tion patterns at various scales of cascading dynamics.

2. Signed cascade model

We propose a model of threshold cascading dynamics

on a signed network by explicitly implementing the role

of negative links preventing the activation of a connected

neighbor. Each node can be one of two states, active or in-

active in multiplicative manner. In signed networks, each

link can be positive or negative. Neighbors connected by

positive (negative) links are referred to as positive (neg-

ative) neighbors for short. A node becomes active when

the following two conditions are fulﬁlled: i) the fraction of

active positive neighbors out of total positive neighbors ex-

ceeds the prescribed threshold θas in the ordinary thresh-

old model [3] and ii) there are no active negative neigh-

bors. The rule clearly shows diﬀerent roles of the positive

and negative connections in the “signed” cascades. While

positive links spread activations to a connected neighbor,

negative links prevents neighbors from activation. Logical

“AND” requirement of both conditions reﬂects the mul-

tiplicative coupling of positive and negative interactions.

According to the model if there exists even one active neg-

ative neighbor, the activation of the node is completely

blocked. We impose the strongest role of negative links in

order to demonstrate the eﬀect of the signed networks in

a simple and dramatic way.

Let us describe the procedures for numerical simulations

of threshold cascading on a signed network. Initially all

nodes are inactive except for a small fraction ρ0of the

seed nodes that are active at the beginning. The signed

cascade proceeds as follows. i) At each step, we select

a node, say i, at random. ii-a) For inactive node i, the

state of node ibecomes active when the ratio of its active

positive neighbors exceeds threshold θand there is no ac-

tive negative neighbor. For instance, suppose that there

are ractive positive neighbors out of kppositive neighbors

for node i. Then node ibecomes active when r/kp> θ

and there is no active negative neighbor. ii-b) If an ac-

tive node including a seed is selected, nothing happens. It

means that active nodes maintain their active state per-

manently. iii) The procedures repeat until the dynamics

of activations reaches a steady state meaning that there

exists no node that can be newly activated. An example

of the signed cascade process is depicted in Fig. 1. We

perform random sequential updates, so that we choose at

random a single node in the network and update its state

at every step. Note that if there is no negative link, our

model reduces to the original Watts threshold model [3].

Contrary to the original Watts threshold model, cascad-

ing dynamics on signed networks is no longer deterministic

because of the role of negative interactions. Speciﬁcally,

the set of active nodes in a steady state can be diverse even

when the cascading dynamics starts from identical seeds

on the same network structure. The ﬁnal conﬁguration

is stochastically realized among many possible conﬁgura-

tions that satisfy the conditions of both activation and

inactivation as illustrated in Fig. 1. The variability of the

ﬁnal conﬁguration depending on the sequence of activa-

tions produces more heterogeneous and richer dynamics

than that of the traditional threshold model.

3. Results

3.1. Suppression of Global Cascades

The primary eﬀect of negative links is the suppression

of cascading dynamics due to the local suppression by an-

tagonistic connections. We examine how negative links

globally suppress cascading dynamics on a signed network

focusing on the global cascades. We measured the size of

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

ThresholdcascadedynamicsonsignedrandomnetworksKyu-MinLeea,b,SungminLeea,ByungjoonMinc,K.-I.GohaaDepartmentofPhysics,KoreaUniversity,Seoul02841,KoreabCollegeofBusiness,KoreaAdvancedInstituteofScienceandTechnology,Seoul02455,KoreacDepartmentofPhysics,ChungbukNationalUniversity,Cheongju28644,KoreaAbstr...

展开>> 收起<<

Threshold cascade dynamics on signed random networks.pdf

共9页,预览2页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Threshold cascade dynamics on signed random networks

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: