Interdependencies and Cascading Effects of Disasters on Critical

Infrastructures: An Analysis of Base Station Communication

Networks

Eva K. Lee

1,2,3 a

and William Zixing Wang

Center for Operations Research in Medicine and Healthcare, The Data and Analytics Innovation Institute, Atlanta, U.S.A.

Georgia Institute of Technology, Atlanta, U.S.A.

Accuhealth Technologies, Atlanta, U.S.A.

Keywords: Critical Infrastructures, Influence Networks, Linear-Threshold Influence Networks, Weighted

Linear-Threshold Influence Networks, Mixed Integer Program, Critical Nodes, Cascading Effects,

Communications Sector, Cellular Base Station Networks, Detecting Maximum Vulnerability, Biological

Intelligence, Airports, Surveillance, Risk, COVID-19.

Abstract: There are sixteen critical infrastructure (CI) sectors whose assets, systems, and networks, whether physical or

virtual, are considered so vital to the United States that their incapacitation or destruction would have a

debilitating effect on military readiness, economic security, public health, or safety. The communications

sector is unique as a critical infrastructure sector due to its central role in facilitating the flow of information,

enabling communication, and supporting all other CIs as well as other components of the economy and society.

Within the communications sector, the cellular base station (cell tower) network serves as its foundational

backbone. During a crisis, if towers in the network stop functioning or are damaged, the service load of

associated users/businesses will have to be transferred to other towers, potentially causing congestion and

cascading effects of overload service outages and vulnerabilities. In this paper, we investigate cellular base

station network vulnerability by uncovering the most critical nodes in the network whose collapse would

trigger extreme cascading effects. We model the cellular base station network via a linear-threshold influence

network, with the objective of maximizing the spread of influence. A two-stage approach is proposed to

determine the set of critical nodes. The first stage clusters the nodes geographically to form a set of sub-

networks. The second stage simulates congestion propagation by solving an influence maximization problem

on each sub-network via a greedy Monte Carlo simulation and a heuristic Simpath algorithm. We also identify

the cascading nodes that could run into failure if critical nodes fail. The results offer policymakers insight into

allocating resources for maximum protection and resiliency against natural disasters or attacks by terrorists

or foreign adversaries. We extend the model to the weighted LT influence network (WLT-IN) and prove that

the associated influence function is monotone and submodular. We also demonstrate an adaptable usage of

WLT-IN for airport risk assessment and biological intelligence of COVID 19 disease spread and its scope of

impact to air transportation, economy, and population health.

1 INTRODUCTION

Critical infrastructures (CI) are systems, assets, and

networks that are essential for the operation of a

country's economy, security, and public health and

safety. They are critical to the functioning and well-

being of a society, and their disruption could have

debilitating effects on private businesses and

government.

https://orcid.org/0000-0003-0415-4640

The importance of CI security and resilience has

been identified by the U.S. government (Presidential

Policy Directive 21 (PPD-21), White House, 2013).

CIs are interdependent – a malfunction in one can

lead to cascading failures in the components of others.

For example, a cyberattack on the power grid could

impact communication networks, transportation

systems and healthcare services. And the most recent

COVID-19 pandemic brings to light the devastating

Lee, E. and Wang, W.

Interdependencies and Cascading Effects of Disasters on Critical Infrastructures: An Analysis of Base Station Communication Networks.

DOI: 10.5220/0012239600003598

In Proceedings of the 15th International Joint Conference on Knowledge Discovery, Knowledge Engineering and Knowledge Management (IC3K 2023) - Volume 1: KDIR, pages 141-152

ISBN: 978-989-758-671-2; ISSN: 2184-3228

141

paralyzing cascading impacts stemming from

healthcare and public health disruptions to the supply-

chain, education, emergency services, food and

agriculture, transportation systems, and government

and commercial facilities. This makes ensuring the

resilience and security of critical infrastructures

crucial for maintaining overall societal functioning.

The communication sector is a unique critical

infrastructure due to its central role in facilitating the

flow of information, enabling communication, and

supporting all the other CIs and other components of

the economy and society. Communication services

are available across diverse environments, including

urban, suburban, and rural areas. They are essential

for connecting individuals and organizations,

regardless of their geographic location. (National

Infrastructure Protection Plan 2015).

Within the communication sector, the cellular

base station network serves as its physical foundation

backbone. During a crisis, if cell towers in the

network stop functioning or are damaged, the service

load of the associated users will have to be transferred

to other nearby towers, potentially causing congestion

and cascading effects of overload service outages and

vulnerabilities.

This paper presents a linear-threshold influence

network to model the cellular base station network.

By maximizing the spread of influence, the system

returns a set of critical nodes that asserts the

maximum cascading effects. Computationally, a two-

stage approach is proposed to investigate the

cascading effects. The first stage involves performing

geographical clustering on all nodes to form sub-

networks. The second stage constructs a linear-

threshold influence network for each sub-network.

The construction of the sub-networks is only

necessary when the problem scale is too large,

rendering the instance computationally intractable.

We also demonstrate its generalizability for

interdependencies and cascading analyses on early

COVID transmission via air transportation.

2 RELATED WORK

Critical infrastructure interdependency was first

investigated in 2001 (Rinaldi et al., 2001). The paper

provides physical, cyber, geographical, and logical

classifications for CI interdependency. Their

subsequent paper summarizes the likely methods for

interdependency analysis (Rinaldi, 2004).

2.1 Modeling Interdependency

Modeling critical infrastructure interdependency is

essential for proactively managing and safeguarding

critical systems that underpin modern societies. It

enables governments, organizations, and

communities to identify and assess vulnerabilities,

enhance resilience, prepare, and train emergency

response, conduct scenario planning, allocate

resources, and develop effective strategies for

maintaining the stability and security of these vital

systems (Atef & Bristow, 2022; Brown et al., 2004;

Delamare et al., 2009; Dudenhoeffer et al., 2006;

Eusgeld et al., 2011; Heracleous et al., 2017; Islam et

al., 2023; Jiwei et al., 2019; Johansson & Hassel,

2010; Lin & Pan, 2022; Nan & Sansavini, 2017;

Ouyang, 2014; Rinaldi, 2004; Santella et al., 2009;

Trucco & Petrenj, 2023; Wang et al., 2022; Yabe et

al., 2022).

There are three methods for modeling

interdependency: simulation-based, analytics-based,

and data-based (Ouyang, 2014; Aung & Watanabe,

2009; Cimellaro et al., 2019; Galbusera et al., 2020;

Robert et al., 2008; Sharma et al., 2021; Tasic et al.,

2019). Among the simulation-based approaches,

Dudenhoeffer designed an agent-based approach to

simulate interdependencies and used a genetic

algorithm to select the CI components to

protect/restore (Dudenhoeffer et al., 2006). In

Johansson and Hassel, the CI interdependency is

modeled as a network and the flows are simulated

when removing edges to find the strains added to the

network (Johansson and Hassel, 2010). Zio and

Sansavini modeled the interdependency as load

transfers where failed nodes would transfer their

loads to adjacent nodes; however, no realistic

experiments were conducted to test how well the

model works (Zio and Sansavini 2011).

Using analytics-based approaches, Wallace et al.

and Lee II et al. modeled the provision

interdependency as a multi-commodity network flow

problem and formulated it as a mixed-integer

program (MIP) (Wallace et al., 2001; Lee II et al.,

2007). Svendsen and Wolthusen designed another

multi-commodity flow formulation but assigned a

response function for each arc and each resource

where some of the resources can be buffered

(Svendsen and Wolthusen, 2007). A drawback of

using network flow problems for this is that it can

only model the provision interdependency and not the

other types.

Data-based models are typically designed based

on specific data forms. For instance, Ramachandran

summarizes the geospatial data to find the CI

KDIR 2023 - 15th International Conference on Knowledge Discovery and Information Retrieval

142

components that would affect most other CI

components geographically (Ramachandran et al.,

2015). Reilly assumes that each CI sector is managed

by a specific governmental or private department and

explores the externality of the policy taken by some

departments as an interdependency (Reilly et al.,

2015).

2.2 Influence Network

Influence networks have proven suitably robust for

investigating the relationships between people and

objects (Calio et al., 2018; Chen et al., 2022; Goyal et

al., 2011; Kempe et al., 2003; Kim et al., 2018; Lee

& Wang, 2017; Palos-Sanchez et al., 2018; Peng et

al., 2018).

The concept of an influence network was first

introduced by Kempe (Kempe et al., 2003). The

authors formulated two influence network models,

proved the submodularity of the influence function,

and the NP-hard complexity of the problem. They

also proposed a Greedy-based Monte-Carlo

algorithm to solve the problem. However, the

network models have two major sources of

inefficiency. First, finding the expected spread of a

node set is NP-hard. Second, the basic greedy

algorithm is quadratic in the number of nodes. In later

years, some researchers attempted to improve the

computational scalability of this approach. Leskovec

discussed the original cost-effective lazy-forward

(CELF) algorithm, and Goyal and Leskovec proposed

CELF++ that utilized the submodular properties of

the influence function to reduce the number of

influence function simulations and the running time

by over 80% (Leskovec et al., 2007; Leskovec et al.,

2009; Goyal et al., 2011). Goyal et al. designed a

faster heuristic solution method known as Simpath to

evaluate the influence function (Goyal et al., 2011).

Simpath does not use Monte-Carlo simulation and

thus improves the running time significantly.

3 MATERIALS AND METHODS

3.1 Modeling the Cellular Base Stations

The modern cellular network consists of several key

components (cellular base stations, mobile devices,

radio access network, core connectivity network,

backhaul network, service providers, and standards

and protocols) that work together to provide the

infrastructure and technology necessary for wireless

communication services. Cellular base stations, also

known as cell towers, are the physical structures that

transmit and receive signals to and from mobile

devices. They are strategically placed to cover

specific areas called cells. Each cell tower is equipped

with antennas and transceivers to communicate with

mobile devices within its coverage range. The cellular

base stations exchange information when

communication is made. When a physical or cyber-

attack paralyzes a cellular base station, users relying

on its coverage need to seek other nearby working

stations, thus asserting extra load burdens to these

locations.

Our model seeks to answer the following

question: Given a number K, which represents the

number of cellular base stations to which our

resources (e.g., additional layer of countermeasures)

can be allocated, determine which K stations, if

attacked, could impact the largest number of stations

in the network. By answering this question, we can

uncover the set of stations that would lead to

maximum protection if strengthened or that would

produce the most severe loss if attacked. Thus, the

extra protective strengthening of this set would lead

to a robust and efficient level of CI security and

resiliency.

Consider a geographical region that is divided

into small cells (calling areas). The modern cellular

network consists of many small calling areas where a

cellular base station serves each area. These base

stations are interconnected using a high-speed

wireless network (Figure 1).

Figure 1: The modern cellular network consists of many

small calling areas where a cellular base station serves each

area. Base stations are interconnected using a high-speed

wireless network.

3.1.1 Choice of Influence Network

Let 𝐺=(𝑉,𝐴) be a directed graph and 𝑆



⊆𝑉 be an

initial node set. Let 𝑁



(𝑣) denote the in-node set of

node v. Two major influence network models have

been proposed in various applications: (Calio et al.,

2018; Chen et al., 2013; Chen et al., 2022; Goyal et

al., 2011; Kempe et al., 2003; Kim et al., 2018; Lee

& Wang, 2017; Palos-Sanchez et al., 2018; Peng et

al., 2018).

Interdependencies and Cascading Effects of Disasters on Critical Infrastructures: An Analysis of Base Station Communication Networks

143

Independent Cascade Influence Network (IC-IN):

Each arc 𝑒∈𝐴 has a probability 𝑝(𝑒). At time 𝑡≥1,

for every inactivated node 𝑣, every 𝑢∈𝑁



(𝑣) ∩

(𝑆



\𝑆



) would try to activate 𝑣 with a probability

𝑝(𝑢,𝑣) independently. The resulting set is denoted by

𝑆



. 𝑣 can be activated only once. 𝑆



=∅.

Linear Threshold Influence Network (LT-IN):

Each arc 𝑒∈𝐴 has a weight 𝑤(𝑒) . For every

inactivated node 𝑣 , it will choose a threshold

𝜃



~𝑈[0,1], where 𝑈 is a uniform distribution. At

time 𝑡≥1, for every inactivated node 𝑣 , if

∑

𝑤(𝑢,𝑣)

∈



()∩



≥𝜃



, 𝑣 is activated. The

resulting set is denoted by 𝑆



. 𝑣 can be activated only

once.

Since the cardinality of 𝑆



is monotonously

increasing and bounded, there exists 𝑆



where

cardinal (𝑆



) = cardinal (𝑆



) and the cardinality

plateaus and does not change anymore. The influence

function σ(𝑆



):𝑆



→𝑅



is defined as the expectation

of the cardinality of the final activated set 𝑆



. For

both IC-IN and LT-IN models, the influence

networks are submodular.

The influence maximization problem is defined as

max σ

(

𝑆



)

, 𝑠.𝑡.

𝑆



=𝐾, where 𝐾 is a given

positive integer. By submodularity, a greedy

algorithm can be used to find the set 𝑆



whose

influence function is at least 1 −

𝑒



 𝑚𝑎𝑥







σ(𝑆



) for a given 𝐾, where 𝑒 is the

base of the natural logarithm (Nemhauser et al. 1978;

Kempe et al. 2003).

In the IC-IN model, each node attempts to

independently activate the adjacent inactivated node,

meaning that for any inactivated node, any activated

node in its in-node set could succeed in activating it.

However, this contradicts the fact that the load

transfer is cumulative in a cellular base station

network. The incoming signal must exceed a certain

threshold (max power) for the base stations to stop

taking new users. Hence the LT-IN model fits the

cellular base station network much better since the

threshold 𝜃



can be interpreted as

  

       

of node

𝑣.

3.2 Learning the Parameters of LT-IN

After choosing the influence network model, the next

step would be defining the network elements.

Naturally, we let each node denote a base station in

the cellular network. To find the arc set A, without

loss of generality, we define a maximum reach

distance 𝑅 for all base stations as the distance from

the most distant user to the base station. For any base

stations 𝑤 and 𝑣, if their distance is less than 2𝑅, we

assume that there might exist a user who initially uses

𝑤 but must use 𝑣 when 𝑤 is paralyzed. Thus, an arc

should go from 𝑤 to 𝑣 and vice versa. For a

heterogeneous design, we can associate a maximum

reach distance 𝑅



for each base station 𝑣.

The final step is to assign weights to the arcs.

Theoretically, for each node 𝑣, let 𝐿





be the maximal

load, 𝐿





be the current load and 𝐿

(,)

be the load

going from 𝑢 to 𝑣 when node 𝑢 is down. Then the

fraction of the residual load for node 𝑣 can be

expressed as

𝜃



𝐿





−𝐿





∑

𝐿

(,)(,)∈

Notice that 𝜃



, as defined, is a U

[

0,1

]

random

variable. This is because the values 𝐿





and 𝐿

(,)

change constantly. Meanwhile, we assume that

∑

𝐿

(,)(,)∈

≥ 𝐿





−𝐿





, which means that if all the

adjacent nodes around 𝑣 are down, 𝑣 would also be

down due to high loads. With these assumptions, we

let the weight on arc

(

𝑢,𝑣

)

𝑤

(

𝑢,𝑣

)

𝐿

(,)

∑

𝐿

(,)

∈



()

so that for any 𝑣 ,

∑

𝑤

(

𝑢,𝑣

)

∈



()

=1. 𝜃



∑

𝑤

(

𝑢,𝑣

)

∈



()

=1 corresponds to the assumption

that 𝑣 is down if all the adjacent nodes around 𝑣 are

down.

Thus, to calculate 𝑤

(

𝑢,𝑣

)

, we only need 𝐿

(,)

In the model, while the geographical positions of the

base stations are known, the users’ locations are

generated uniformly across the target area. For each

user, we associate it with the nearest station as its

primary base station. In addition, we also assign the

second nearest station for each user as the station that

the user would connect to when its primary station is

down. In this way, 𝐿

(,)

is the number of users who

use 𝑢 as their primary station and 𝑣 as their

secondary station. For 𝑤

(

𝑢,𝑣

)

, we are not

concerned with the absolute value of 𝐿

(,)

, but rather

the percentage it represents within

∑

𝐿

(,)

∈



()

Thus, the number of virtual users does not matter as

long as it is sufficiently large.

3.3 Two-Stage Framework to Analyze

the Influence Network

When base station A is down, its associated users

must turn to other nearby stations, which will increase

KDIR 2023 - 15th International Conference on Knowledge Discovery and Information Retrieval

144

the burden on those base stations. Although the

maximum power for the base stations nowadays is

high, when multiple base stations nearby are down, it

is still possible that the load becomes dangerously

high. Suppose that base station B takes many users

associated with broken base stations and reaches its

maximum capacity load. New users must turn to other

base stations except A or B, causing cascading effects.

Given a network of base stations, if there are some

resources and countermeasures that can be used and

taken to improve the protection of

stations from

potential attacks (wireless or physical) which

stations are the most critical.

We are interested in uncovering the set of

base

stations that maximizes the cascading of influence on

the overall network (i.e., the cascading effect on the

maximum number of base stations). One major hurdle

in influence network modelling involves

computational challenge. For large networks, the

MC-Greedy method requires a long running time to

solve. As for the Simpath method, although it is

scalable, it does not have a theoretical lower bound

and often performs poorly on large and complex

networks. For this study, we design a two-stage

framework using these two algorithms (along with

submodularity to reduce the number of iterations) to

analyze the cellular base station problem.

Stage 1. Forming Sub-Networks by Clustering

To speed up the solution time, we partition the large

complex network into smaller sub-networks. Since

the influences cannot spread over long ranges, we use

the K-means++ clustering method (Arthur and

Vassilvitskii, 2007) to cluster geographically. Once

the nodes are clustered, the arcs (interdependencies)

between nodes that belong to different clusters will be

removed.

Stage 2. Optimizing to Uncover the Most

Influential Node-Set

Suppose Stage 1 returns 𝑛 sub-networks. Given a

positive integer

, we want to determine the set of

critical nodes from the entire network that asserts the

maximum influence. In the simplest approach, we

solve this problem on each sub-network by

uncovering 





 or 





 critical nodes that asserts the

maximum influence. While one can estimate the

number of critical nodes that need to be selected

based on the size of each sub-network, the simple

decomposition could ensure that the final node-set

distribution is generally uniform across the target

area. It is worth noting that the choice of 𝑛 implies a

trade-off between fewer artificial restrictions on the

original network versus higher solution precision and

faster solution time on each sub-network. Policy

makers should experiment with various choices of 𝑛

to contrast the quality and practical implications of

the resulting solutions.

4 EXPERIMENTS AND

SENSITIVITY ANALYSIS

4.1 Data and Experiments

We test our model using the cellular station data set

on Homeland Infrastructure Foundation-Level

databases provided by the U.S. Department of

Homeland Security (DHS Cellular tower data, 2018).

The dataset includes the geographical locations of

23,498 cell towers in the United States.

We set the maximum reach distance 𝑅 = 5 km to

formulate the LT-IN for the cellular base station

network. When applying the two-stage framework to

analyze the LT-IN, we choose 𝐾=100, i.e., we want

to uncover the 100 most influential nodes that assert

maximum influence across the entire dataset of

cellular base stations. We implement in-house

Simpath and MC-Greedy algorithms in Python. For

network partitioning, we choose the grid size to be

factors of 10 to generate the number of clusters n. We

also attempt the scenarios where 𝑛 equals 25 and 33

(they lead to 4 and 3 critical nodes in each cluster).

This results in the number of clusters with n = [10,

20, 25, 33, 40, 50, 60, 70, 80, 90, 100] for

experimentation. In the MC-Greedy approach, every

MC simulation is set for 1000 runs.

The results are presented in Figures 2 and 3.

Figure 2 plots the final number of influenced/affected

nodes against various cluster values 𝑛. For each

algorithm, when 𝑛 increases, the number of final

affected nodes is not monotone, but contains several

turning points. For Simpath, the maximal number

appears at 𝑛=33; for MC-Greedy it appears at 𝑛=

60.

When the number of clusters is less than 40,

Simpath returns solutions that assert more influence

than the solutions from MC-Greedy, while MC-

Greedy outperforms Simpath when the number of

clusters is over 40. This is potentially because when

the sub-network is large (i.e., a small value of n), the

number of critical nodes to be chosen in each cluster

is large; as a result, the MC-Greedy simulation

requires more simulation rounds to be accurate, thus

returning poor results. In general, the number of

Interdependencies and Cascading Effects of Disasters on Critical Infrastructures: An Analysis of Base Station Communication Networks

145

influenced nodes obtained by Simpath lies within [-

7%, +11%] of those by MC-Greedy.

Figure 2: The number of final affected nodes when

choosing 100 base stations.

Figure 3: Running time to uncover the 100 most influential

base stations.

Figure 3 shows the running time for the two

approaches. As a scalable and heuristic approach,

Simpath is much faster for all values of n.

Specifically, Simpath manages to solve the entire

network in 8,704 CPU seconds; it requires 341 CPU

seconds when n = 10, and under 150 CPU seconds for

all other instances. We can observe that the

computational performance of MC-Greedy worsens

(i.e., requires a longer solution time) when the

number of nodes in each cluster increases due to

interdependencies in the network. It fails to solve any

instances when n < 10 and requires over 100 CPU

hours to solve when n = 10.

4.2 Sensitivity Analysis

Similarities Between Solutions Obtained by the

Two Methods

After obtaining the results from both algorithms, a

natural question is to examine whether they return

similar results, e.g., how many of the chosen stations

overlap? Table 1 shows that on average about 20% of

the chosen nodes overlapped.

Table 1: Number of overlapped critical nodes across two

methods.

Number of Clusters Overlapped Critical Nodes

10 23

20 15

25 14

33 13

40 15

50 24

60 20

70 16

80 18

90 23

100 22

We next examine how close geographically the

non-overlapping nodes are. Figures 4, 5, and 6

compare the nodes chosen by the two methods when

the number of clusters are 33, 50, and 100,

respectively. From the figures, it is observed that

when there are 100 clusters (i.e., one critical node is

chosen), the chosen nodes by the two methods are not

geographically close. On the other hand, for the 50

and 33 clusters, even though the number of

overlapping nodes is not much higher than 22% (24%

and 13% respectively) as in the 100 clusters, the non-

overlapping nodes chosen by the two methods are

rather close to each other.

Figure 4: Critical nodes chosen by two methods when the

base stations are partitioned into 100 clusters.

200

400

600

0 20406080100

Number of nodes affected

Number of clusters

Number of final affected/influenced nodes

by choosing 100 critical stations

MC-Greedy

Simpath

50000

100000

150000

200000

250000

300000

350000

400000

0 20406080100

Run time (CPU seconds)

Number of clusters

Run time to uncover 100 stations

MC-Greedy

Simpath

KDIR 2023 - 15th International Conference on Knowledge Discovery and Information Retrieval

146

Figure 5: Critical nodes chosen by two methods when the

base stations are partitioned into 50 clusters.

Figure 6: Critical nodes chosen by two methods when the

base stations are partitioned into 33 clusters.

Similarities Among Different Partitioning

We are interested in learning how critical nodes are

selected among different partitions. Figure 7 and 8

contrast critical nodes obtained via the 33, 50, and

100 clusters using the MC-Greedy algorithm and

Simpath algorithm respectively. We observe that for

both methods, the critical nodes selected from 33 and

50 clusters are close to each other, while the nodes

from the 100 clusters are further apart.

Figure 7: Critical nodes chosen by the 33, 50, and 100

clusters respectively, obtained by the MC-Greedy

algorithm.

Figure 8: Critical nodes chosen by the 33, 50, and 100

clusters respectively, obtained by the Simpath algorithm.

5 WEIGHTED

LINEAR-THRESHOLD

INFLUENCE NETWORK

5.1 Investigating Risks/Biological

Intelligence

Airport Risk Assessment

We illustrate the generalizability of our model by

reporting a prospective analysis carried out during the

early period of COVID-19 in March 2020.

Herein, our single-layer influence model was

applied to the air transportation component of the

transportation CI sector. We are interested in

identifying the set of critical airports that maximize

risk and the associated cascading effects on

transportation, the population, and the economy.

From U.S. Census Bureau data, the 12 major

metropolitan areas include New York, Los Angeles,

Chicago, Dallas, Houston, San Francisco-San Jose,

Washington D.C., Miami, Atlanta, Philadelphia,

Boston, and Seattle. And the five minor metropolitan

areas which have population over three million are

Phoenix, Detroit, Minneapolis, San Diego, and

Tampa.

Let matrix F denote the number of flights between

each pair of 216 major U.S. airports.

Let W denote the initial weights (216*1). The

airport weights in the network are calculated by 𝑊∙

(

𝐹𝑊

)

, where the operation “∙” represents the

element-wise product.

FW is the weighted sum of outbound flows for

each airport, and W∙ (FW) couples the origin airport

weights to the outbound flows.

Herein, we contrast results with W chosen in three

different ways:

Interdependencies and Cascading Effects of Disasters on Critical Infrastructures: An Analysis of Base Station Communication Networks

147

: Uniform weights: W = 1, W∙ (FW) = F, the

unweighted sum of flow.

: Weight by population: For major

metropolitan areas, w = 3, for minor metropolitan

areas, w = 2 and for others, w = 1.

: Weight by annual average daily volume of

each airport.

Table 2: Critical airport selected when K = 5 (blue), 10

(blue+red), and 15 (all three colors), respectively.

MC-

Greedy Simpath

MC-

Greedy

MC-

Greedy Simpath

MC-

Greedy

LAS X X XX X X

PHX X X XX X X

DEN X X XX X X

ATL X X XX X X

MCO X X X X X X

LAX X X X X X X

ORD X X X X X X

FLL X X X X X X

DFW X X X X X

OGG X

SFB X X X

TPA X XX X

MSP X X XX X X

DTW X X XX X X

SEA X X X X X

BWI X X X X

SFO X X

X X X X

STT X

DAL X

Table 2 shows the selected critical airports when

K = 5, 10, and 15 and under different weights. Blue

indicates the first selection of five; Red is the next five

and black is subsequently five more. We observe that

the two approaches are quite consistent in selecting

the airports and that the weights offer some variants

in the resulting solutions. For example, when

population (W

) and annual daily volume (W

)

weights are involved, multiple international airports

are selected from Florida (MCO, FLL, SFB, TPA)

due to its theme parks and extensive cruise ports. The

cascading effect here corresponds to the impact on the

transportation functions, economy, and local

population. These critical nodes can offer guidelines

on countermeasure allocations to various airports for

maximum protection. It also reflects the

vulnerabilities they face. Hence our solutions offer

both defensive and offensive knowledge.

Biological Intelligence on COVID Spread

During COVID-19, we used the airport results

obtained in 2019 to prospectively validate the

vulnerabilities of disease cases identified at the

airports. In the early stage of COVID-19 in the U.S.,

on March 6, 2020, we had the following facts:

 *LAS (Las Vegas), *PHX (Phoenix). *DEN

(Denver), *ATL (Atlanta), *MCO (Orlando)

 *LAX (Los Angeles), *ORD (Chicago), *FLL

(Fort Lauderdale), DFW (Dallas), OGG

(Hawaii)

 *MSP (Minneapolis), DTW (Detroit), *SEA

(Seattle), *BWI (Baltimore-Washington), SFO

(San Francisco), *JFK (New York).

The pink star denotes airports with confirmed

cases. The blue star represents reported travellers

with direct contact to a confirmed COVID-19

individual not allowed to board the plane back to the

U.S. that self-quarantined but was not tested. San

Francisco had COVID-19 at the time but it was not

reported to be travel-related. Dallas and Hawaii did

not have reported air travellers with COVID-19 at the

time.

This illustrates that our model, which features

maximum influence optimization, is flexible and

adaptable for modeling the interdependencies and

connectivity of CIs and provides a good systems-risk

framework for a broad spectrum of scenario

predictions. Early intervention could include

prioritization of diagnostic test resources or self-

quarantine recommendations at those critical airports.

From this airport analysis, we can observe that the

LT-IN influence maximization problem is very

adaptable to a diverse type of CIs and their

components, rather than specific to the functioning

nature of one particular CI. Such a modeling construct

is very appealing; however, it remains important to

set up the model properly with meaningful arcs and

parameters for interpretable and insightful outcomes.

The airport analyses utilize the concept of

weighted linear-threshold influence network, which

we formally introduce in Section 5.2. We also prove

that the weighted influence function is monotone and

submodular.

5.2 Submodularity in Weighted LT

Influence Network

Weighted LT Influence Network: The graph

G=(V,A,W) has weights 𝑤

(

𝑒

)

∈ (0,1] for each arc

𝑒∈𝐴 and positive weights 𝑊(𝑣) for each node 𝑣∈

𝑉 . For every inactivated node 𝑣, it will choose a

threshold 𝜃



~𝑈[0,1] , where 𝑈 is a uniform

distribution. Let 𝑁



(𝑣) denote the in-node set for

node 𝑣. At time 𝑡≥1, for every inactivated node 𝑣,

∑

𝑤(𝑢,𝑣)

∈



()∩



≥𝜃



, 𝑣 is activated. The

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148

resulting set is denoted by 𝑆



. 𝑣 can be activated only

once.

The influence function 𝑓(𝑆



):𝑆



→𝑅



, is

defined as 𝐸(

∑

𝑊(𝑣))

∈



when the initial active set

is 𝑆



, where 𝐸() is the mathematical expectation.

𝑆



exists since 𝑆



is monotonically increasing

and bounded.

The influence maximization problem is

max 𝑓

(

𝑆



)

, 𝑠.𝑡.

𝑆



=𝐾, where 𝐾 is a given

positive integer.

Theorem: The influence function 𝑓() in the

weighted LT influence network is monotone and

submodular.

Proof: The live-arc graph for regular LT influence

network is applied here. Let L denote the set of all

live-arc graphs of this network. The influence

function for the initial set 𝑆



can be represented as:

𝑓

(

𝑆



)

=𝑃(





∈

𝐺



)𝑊(𝑣)

∈





(



)

where 𝑅





(𝑆



) is the node set connected to the initial

set in live-arc graph 𝐺



We know that the linear combination of

monotone (resp. submodular) functions with non-

negative coefficients is also monotone (resp.

submodular). It is sufficient to show that for any live-

arc graph 𝐺



∑

𝑊(𝑣)

∈





(



)

is monotone (resp.

submodular) w.r.t 𝑆



Monotonicity is trivial so we will only prove

submodularity. By theorem 2.13 in Chen et al. (2013),

for any 𝑆⊆𝑇⊆𝑉 and 𝑢∈𝑉\𝑇, we have:

𝑅





(𝑇 ∪ {𝑢})\𝑅





(𝑇) ⊆ 𝑅





(𝑆 ∪ {𝑢})\𝑅





(𝑆)

Since 𝑊(𝑣) is positive, we have:

𝑊(𝑣)

∈





(∪{})

−𝑊

(

𝑣

)

≤

∈





()

 𝑊

(

𝑣

)

−𝑊(𝑣)

∈





()∈





(∪{})

which means

∑

𝑊

(

𝑣

)

∈





(





)

is submodular and

thus 𝑓

(

𝑆



)

is submodular.

6 CONCLUSIONS

Critical infrastructures are fundamental facilities and

services that are necessary for the functioning of a

society and its economy. Analyzing the

interdependency and cascading effect in critical

infrastructure is crucial for building a more resilient,

secure, and sustainable society. It enables better

planning, risk management and response efforts to

safeguard essential services and ensure the continuity

of daily life, especially in the face of numerous

modern challenges and emergencies.

In this paper, we present a method to model

interdependency and cascading effect for critical

infrastructures. Our model utilizes the linear-

threshold influence network and influence

maximization to determine critical nodes in the CI

that have the most influence when disrupted. We

designed a two-stage framework to analyze the LT-

IN.

Applying it to the cellular base station network

within the U.S., the results identify the optimal

partition in the network. The two algorithms, MC-

Greedy and Simpath, enable comparison, sensitivity

analysis and cross-referencing on the solution quality.

The critical nodes and their influence correspond to

the most connected / inter-dependent node structure

in the entire network. Such knowledge sheds light on

the network’s vulnerabilities and enables the

development of effective resilience plans. By

identifying areas with the greatest potential impact

and vulnerabilities, it offers a good reference for

policy makers on how to allocate (limited) resource

strategically to ensure that investments are made

where they are most needed, and that it protects the

overall communication infrastructure most

efficiently.

During a crisis or disaster, results from our model

can provide a basis for predicting how events would

unfold and disrupt and impact others, helping

authorities to understand trade-offs and make quick

and informed decisions. Governments and regulatory

bodies can work with communication sector

businesses to develop policies and regulations that

ensure the security and stability of communication

CI. For example, the models can inform regulations

regarding scope of disruption and cost-effectiveness

of redundancy plans in secondary/backup base station

assignment, or measures to mitigate the impact of

disruptions. Users can explore various scenarios and

"what-if" analyses, which can be beneficial for

understanding the consequences of different types of

disruptions and planning accordingly.

We extend the LT-IN model to the weighted LT-

IN model and prove that the associated influence

function is monotone and submodular. Applying it to

airport risk influence maximization analysis

demonstrates the diverse usage of the WLT-IN model

Interdependencies and Cascading Effects of Disasters on Critical Infrastructures: An Analysis of Base Station Communication Networks

149

in risk assessment and the evaluation of potential

consequences. For COVID-19, it reflects disease

spread and its scope of impact to transportation, the

economy, and population health. The critical nodes

and their manifesting influence can help decision

makers prioritize resources and investments to

address the most critical vulnerabilities and reduce

the overall impact of potential incidents. For example,

public health officials can establish guidelines on

diagnostic testing and quarantining strategies based

on airport vulnerabilities and overall cascading

impact. Understanding interdependencies can help

businesses and government to develop effective

continuity plans and contingencies to minimize

disruptions during crises.

Because we can and have identified these critical

airports before disasters/pandemics strike, plan-ahead

operations can be carried out for more effective

containment. Furthermore, when the cascading

disruption is better understood and communicated to

the public, it can lead to increased awareness and

preparedness among individuals and communities.

This can be particularly important for disaster

planning and response.

Currently, we are extending the network

formulation and analysis framework to other critical

infrastructures. We are also developing a multi-layer

influence network model to analyze the

interdependencies across multiple CI sectors

(preliminary results reported in Chapter 5 in Wang,

2020).

ACKNOWLEDGEMENTS

This material is based upon work supported by the

U.S. Department of Homeland Security under Grant

Award Number 17STQAC00001-01 in which EK

Lee served as the principal investigator for the project

entitled “Interdependencies and Cascading Effects of

Disasters on Critical Infrastructure”. The views and

conclusions presented in this document are those of

the authors and should not be interpreted as

necessarily representing the official policies, either

expressed or implied, of the U.S. Department of

Homeland Security.

The LT-IN model and its application to the

cellular base stations were first presented at the U.S.

Department of Homeland Security Centers of

Excellence Summit 2018. A more detailed

description can be found in Chapter four of ZX

Wang’s PhD thesis (Wang, 2020). Airport biological

intelligence was analyzed and discussed by EK Lee

to the multi-agency Red Dawn COVID collaborative

in March 2020.

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