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Influence spread in twolayer interdependent networks: designed singlelayer or random twolayer initial spreaders?
Applied Network Science volume 4, Article number: 40 (2019)
Abstract
Influence spread in multilayer interdependent networks (MIDN) has been studied in the last few years; however, prior works mostly focused on the spread that is initiated in a single layer of an MIDN. In real world scenarios, influence spread can happen concurrently among many or all components making up the topology of an MIDN. This paper investigates the effectiveness of different influence spread strategies in MIDNs by providing a comprehensive analysis of the time evolution of influence propagation given different initial spreader strategies. For this study we consider a twolayer interdependent network and a general probabilistic threshold influence spread model to evaluate the evolution of influence spread over time. For a given coupling scenario, we tested multiple interdependent topologies, composed of layers A and B, against four cases of initial spreader selection: (1) random initial spreaders in A, (2) random initial spreaders in both A and B, (3) targeted initial spreaders using degree centrality in A, and (4) targeted initial spreaders using degree centrality in both A and B. Our results indicate that the effectiveness of influence spread highly depends on network topologies, the way they are coupled, and our knowledge of the network structure — thus an initial spread starting in only A can be as effective as initial spread starting in both A and B concurrently. Similarly, random initial spread in multiple layers of an interdependent system can be more severe than a comparable initial spread in a single layer. Our results can be easily extended to different types of event propagation in multilayer interdependent networks such as information/misinformation propagation in online social networks, disease propagation in offline social networks, and failure/attack propagation in cyberphysical systems.
Introduction
Multilayer interdependent networks (MIDNs) are systems composed of more than one network with edges between them to form an interconnected environment. MIDNs can be used to model many realworld interdependent systems, such as cyberphysical systems and onlineoffline social networks. In recent years, we have seen an exponential growth in the development and deployment of these systems. This growth is largely due to the increasing use of more technologies interfacing with one another — such as the Internet of Things. The explosive expansion and scale of technologies that connect with one another through the Internet provide further incentive to explore the interconnection of different realworld systems and their impacts on phenomena propagation in these environments.
The interdependency between separate networks creates specific characteristics for these systems that need to be investigated. Specifically, the interaction between separate networks opens up potential for unique opportunities to design new selection strategies that might not be as effective in a system made up of a single network. For example, the attack that was launched by Stuxnet (Albright et al. 2010) in 2010 was the result of a malicious computer worm that led to substantial damages to the programmable logic controller (PLC) systems and the power plants. The main reason for such widespread damage was the interdependency between the supervisory control and data acquisition (SCADA) system controlling the nuclear enrichment plants and the enrichment plants. This huge impact would likely not have been possible without the interdependency between the cyber (controller) component and the physical component’s performance.
Recently, research aimed at investigating widespread phenomena propagation in MIDNs focuses on designing resilient and robust MIDN systems. There is a considerable amount of literature addressing the problem of modeling and analyzing the impact of interdependency between different interdependent systems (such as cyberphysical systems) and how to minimize phenomena propagation in these systems. Work investigating selection strategies of initial spreaders in MIDN systems has been conducted (as discussed in Related Work section). However, these works consider less general propagation models like epidemic and independent cascade. Additionally, our work is particularly interested in investigating how the evolution of affliction over time and a variety of interconnectivity and layerbased selection strategies for initial spreaders compare, distinguishing our work from that in the literature.
In short, we will make the following contributions:

We propose a modified version of the thresholdbased phenomena propagation model that was initially proposed in (Khamfroush et al. 2016) for a general interdependent network and a single phenomenon, as a mathematical model to quantify the impact of different strategies for selecting initial spreaders for phenomena propagation in an MIDN system.

We perform extensive simulations to study the effectiveness of different types of spread initialization including random/designed singlelayer initial spread and random/designed multilayer initial spread in a MIDN system.

We provide guidelines on which network topologies and types of coupling between the networks provide faster propagation when facing different strategies for spread initialization. Due to the generality of the proposed model, our observations provide a useful framework for further investigation and design of robust and reliable MIDN.
Related work
Various works explore different approaches to either minimizing or maximizing phenomena propagation in MIDN systems, depending on the context of the problem. For instance, in social networks, one may want to maximize the spread of critical information (Kim et al. 2014; 2015; Kandhway and Kuri 2017) or minimize the propagation of misinformation (or “fake news”) (Papanastasiou 2018; Kimura et al. 2009; Shu et al. 2019). Since our problem is general in nature, we do not consider a context that implies whether the phenomena propagation has positive or negative ramifications. We are simply interested in how certain selection strategies of initial spreaders affect overall phenomena propagation.
Securing an MIDN goes beyond securing the separate networks composing the entire interdependent topology. Adversaries are willing to use the interdependency of vulnerabilities to carry out multistage attacks. Each facet of an attack may not pose a significant threat to a single network on its own; however, cumulative influences through interdependency, may have catastrophic effects.
In the past few years, vulnerabilities in MIDNs have been widely studied. In general, there are two main approaches to study this vulnerability. The first approach typically involves focusing on a certain application of MIDNs to identify different forms of attacks/threats and potential strategies to protect corresponding MIDN systems. For example, (Anderson and Fuloria 2010; McDaniel and McLaughlin 2009; Metke and Ekl 2010; Mo et al. 2012; Rahman et al. 2012) looked into financially motivated threats in smart grids where a customer who wants to trick a utility company’s billing system tampers with smart meters to reduce the electricity bill. Other studies, such as (Halperin et al. 2008; Hanna et al. 2011; Rushanan et al. 2014), looked into the threats against medical cyberphysical systems. Authors in (Brooks et al. 2008; Checkoway et al. 2011; Hoppe et al. 2008) defined and investigated different threats in smart cars as another application of cyberphysical systems.
In our prior work (Hudson et al. 2019), we investigate the effectiveness of standard and interdependent centrality metrics to minimize cascade of failure. In that work, we used random failure to select initially failed nodes to begin failure propagation. There was no investigation, in this prior work, into how different strategies for initial spread performed.
In the past, there have been studies conducted that investigate the effectiveness of selection strategies in singular networks (Albert et al. 2000; PastorSatorras and Vespignani 2001; Cohen et al. 2001). However, authors in (Salehi et al. 2015) provide a thorough overview of prior works investigating this topic for MIDN networks. These prior works consider different models of phenomena propagation for the spread of a phenomenon through an MIDN system. For example, (Erlandsson et al. 2017) investigates seed selection strategies to maximize information cascade in multilayer social networks under an independent cascade model (ICM), where nodes across different layers are one and the same — so if a node i_{A} is afflicted in layer A from an intraneighbor in that layer, node i_{B} in layer B is afflicted as an immediate result. This work concluded that degree centrality, out of the strategies considered by this work, is the overall most effective selection strategy for initial spreaders to maximize cascade in these environments. However, this work does not consider propagation over time as afflicted nodes only have one chance to afflict their neighbors. The work in (Zhao et al. 2014) considers an epidemic propagation model, where the goal is to identify the most influential nodes in the network. Also in the context of social networks, there have been a large body of work investigating the influence maximization problem that focuses on choosing optimal seed set such that the spread of information is maximized, e.g., (Michalski et al. 2014; Kempe et al. 2005; Chen et al. 2009; Chen et al. 2010).
In contrast to these prior works, our goal is not to find the most influential seed set, instead we are looking at the impact that different choices of the seed set (singlelayer versus twolayer) could have on the propagation process. More specifically, we are investigating whether a singlelayer seed selection could be as influential as a similar size twolayer seed selection. Furthermore, our study incorporates a more general propagation model and takes into consideration the evolution of the spread of a phenomenon over time. This consideration allows afflicted nodes more opportunities to afflict neighbor nodes, which is more realistic in many scenarios. For instance, in the context of information cascade in social networks one user may be afflicted by some information and share that with their friends. This person’s friends may not immediately adopt the information that this person is propagating; however later in time this could change. The independent cascade model does not consider such cases and is thus inapplicable for many scenarios that involve delay. Further, linear threshold and linear probabilistic propagation are special use cases of our model.
Problem statement
We consider an MIDN system consisting of two interdependent networks, A and B. Both layers A and B are represented by two undirected graphs G_{A}=(V_{A},E_{A}) and G_{B}=(V_{B},E_{B}), where V_{A} and V_{B} represent the sets of vertices in layers A and B (respectively) and E_{A} and E_{B} represent the sets of edges in layers A and B (respectively). It is assumed that V_{A}=N_{A} and V_{B}=N_{B}. Though it is not contingent for N_{A} to be equal to N_{B}, for our experiments N_{A}=N_{B} in all synthetic cases. The two networks are interconnected by means of directed edges. We refer to edges that connect nodes belonging to the same networks as intraedges and those that connect nodes belonging to different networks as interedges. We use directed edges for interedges to capture different models of interdependency between networks. Without loss of generality, it is assumed that an initial set \(F_{0}=F_{0}^{A} \cup F_{0}^{B}\) of nodes are afflicted initially, where \(F_{0}^{A}\) represents the set of initial spreaders in network A and \(F_{0}^{B}\) represents the set of initial spreaders in layer B. Phenomena can propagate among nodes belonging to the same network. In addition, phenomena can also propagate across networks, such as phenomena propagating from a node in layer A to a node in layer B (or viceversa) through interedges. Phenomena propagation takes place with respect to a thresholdbased propagation model — as discussed in more detail in “Phenomena propagation model” section. Given an MIDN system with a known topology, we are interested in comparing the rate of phenomena propagation with respect to time under different types of initial spreads. More specifically, given an initial set of afflicted nodes F_{0} (“initial spreaders”), we will evaluate phenomena propagation in an MIDN system. The goal of this work is to provide useful guidelines on how to design reliable MIDN systems. Additionally, we are particularly interested in the evolution of phenomena propagation in the case of single and twolayer affliction.
To model different types of initial spreads in MIDN systems, we will define different values for sets \(F_{0}^{A}\) and \(F_{0}^{B}\) and different selection strategies for these sets. More specifically, we analyze the following four types of initial spreads in an MIDN.

Random singlelayer spread is represented by defining \(F_{0}^{A}=F_{0}\) and \(F_{0}^{B}=0\), where the nodes of \(F_{0}^{A}\) are chosen randomly^{Footnote 1}.

Random twolayer spread is represented by defining \(F_{0}^{A}=F_{0}/2\) and \(F_{0}^{B}=F_{0}/2\), where the nodes of \(F_{0}^{A}\) and \(F_{0}^{B}\) are chosen randomly.

Designed singlelayer spread is represented by choosing F_{0} nodes belonging to layer A with highest degree \(F_{0}^{A}\), and setting \(F_{0}^{B}=0\).

Designed twolayer spread is represented by choosing F_{0}/2 nodes belonging to layer A with highest degree \(F_{0}^{A}\) and F_{0}/2 nodes belonging to layer B with highest degree \(F_{0}^{B}\).
Due to the general nature of our problem, we are not seeking to differentiate negative and positive phenomena propagation. We are simply interested in how certain selection strategies for initial spreaders affect overall phenomena propagation. For this reason, we refer to nodes that have been affected by the phenomena propagation as “afflicted” and the process by which nodes are afflicted as “affliction”. This language is used to avoid the contextual effects of phenomena propagation for different domains. For an overview of the notation defined throughout this paper, refer to Table 1.
Phenomena propagation model
Many thresholdbased propagation models rely on deterministic behavior; wherein if a proportion of a node’s neighbors are afflicted, then it will be afflicted definitively. While this approach is appropriate for epidemic phenomena, this approach to phenomena propagation is not as general and fails to consider cases with probabilistic propagation. Due to this limitation we incorporate a modified version of probabilistic thresholdbased phenomena propagation used in our prior works (Khamfroush et al. 2016; Hudson et al. 2019). As in our prior works, we considered nodes belonging to the same network to have peer roles and thus are connected through undirected edges. We refer to the adjacency matrices that represent layers A and B as \(M_{AA} \in \{0,1\}^{N_{A} \times N_{A}}\) and \(M_{BB} \in \{0,1\}^{N_{B} \times N_{B}}\), respectively. The interconnectivity between layers A and B can be represented as interconnection adjacency matrices \(M_{AB} \in \{0,1\}^{N_{A} \times N_{B}}\) and \(M_{BA} \in \{0,1\}^{N_{B} \times N_{A}}\) — with the former representing directed edges from layer A to layer B and the latter representing the opposite.
We consider two nodes belonging to the same network with edges between each other as intraneighbors. Additionally, we consider a node i with a directed edge to a node j that belongs to a different network as an interparent. Formally, the set of intraneighbors for node i in some layer A can be defined as \(U_{A}^{intra}(i)=\{j \in A \mid (i,j) \in E_{A}\}\), with \(U_{A}^{intra}(i)\) representing the intradegree of node i∈A. Further, we formally define the set of interparents belonging to some layer B of a node i belonging to some layer A as \(U_{A}^{inter}(i)=\{j \in B \mid M_{BA}(j,i)=1\}\).
Under this model, a node has a probability that it will become afflicted only under the case that the proportion of its afflicted neighbors reach or exceed the designated threshold. This model relies on specified a p_{max} value for each set of connectivity between networks — for our work, we consider p_{max(AA)}, p_{max(AB)}, p_{max(BB)}, and p_{max(BA)}. This is a general parametric model that incorporates many previous models as special cases. For the two networks composing the MIDN topology for our study, we introduce two different threshold functions: k_{aa}(i)∈(0,1] for i∈A, and k_{bb}(i)∈(0,1] for i∈B to model the propagation across the nodes of the same network. We also introduce two other threshold functions: k_{ab}(i)∈(0,1] for i∈B, and k_{ba}(i)∈(0,1] for i∈A, to model propagation across the two interdependent networks, from layer A to layer B and vice versa, respectively. We denote π(i) to be the probability of phenomena propagation to afflict node i due to the propagation of intraneighbors of i, and π^{′}(i) to be the probability of phenomena propagation to afflict node i due to the propagation of interparents of i.
Assuming that a fraction α(i) of intraneighbors of node i are already afflicted, the probability that phenomena propagates to node i within one timestep is defined in Eq. 1,
where p_{max(AA)}(i) represents the probability that node i is afflicted within one time step, when all of its intraneighbors are already afflicted. Similarly, node i may become afflicted due to the propagation of some of its interparents. Let β(i) be the fraction of interparents of a node i that are already afflicted. Thus, the probability that phenomena propagates from interparents of node i to node i within one time step is calculated as defined in Eq. 2,
where p_{max(BA)} represents the probability that node i in layer A is afflicted within one time step, when all of its interparents in B are already afflicted. We use a similar notation for node j∈B by defining equations analogous to Eqs. 1 and (2), where we use the thresholds k_{BB} and k_{AB} to express the required fraction of intraneighbors and interparents of node j that must be afflicted before j can become afflicted with positive probability. Also, the probability that node j in layer B is afflicted within one time step, when all of its interparents in A are already afflicted will be shown by p_{max(AB)}. Note that p_{max(AB)} and p_{max(BA)} are not necessarily equal.
Generalization for many layers
This model can easily be extended to consider MIDN topologies consisting of some abstract m layers. In this work, we only consider MIDN topologies with two layers, so we provide notation to understand the formal definitions of this model with respect to that (e.g., M_{AA}, p_{max(AB)}, etc.). However, this model can also be used to consider MIDN topologies made of more than two layers. For instance, in an MIDN topology of three layers (A, B, and C), you would consider all the same formal definitions, but include parameter permutations that involve C — such as M_{CC}, M_{CA}, M_{AC}, M_{CB}, M_{BC}, etc.
The decision to solely focus on MIDN topologies of two layers for this work was motivated by the curiosity to explore how varying interconnectivity, selection strategies, sizes, and other parameters affect the evolution over time of the phenomena propagation process. More specifically, we would like to study to what extent interdependency between networks and the knowledge of network topology can help the speed of propagation in a MIDN network.
Phenomena propagation process
The temporal evolution of phenomena propagation is modeled as a Markov model. This is because the next state of the network will only depend on the current state of the network and it is independent of how the network reaches its current state. In the following, we describe the elements of our proposed Markov model.
State definition
We denote by S_{T} the set of all possible states of the model, where each state is defined as a vector \(s=(\overbrace {s_{1},s_{2},\dots,s_{N_{A}}}^{{A}},\overbrace {s_{N_{A}+1},s_{N_{A}+2},\dots,s_{N_{A}+N_{B}}}^{B})\), where s_{i}=1 for i≤N_{A} if node i∈A is afflicted, and s_{i}=0 if it is not afflicted. Similarly, for i>N_{A}, s_{i}=1 is one if node i−N_{A} of layer B is afflicted, and s_{i}=0 otherwise. Therefore, the initial state of the phenomena propagation process is \(\phantom {\dot {i}\!}S_{0}=(s_{1},s_{2},\ldots,s_{N_{A}},s_{N_{A}+1},s_{N_{A}+2},\ldots,s_{N_{A}+N_{B}})\), where s_{k}=1 if \(k \leq N_{A} \wedge k \in A \cap F_{0}^{A}\) or \(k >N_{A} \wedge k \in B \cap F_{0}^{B}\), while s_{k}=0 otherwise. \(F_{0}^{A}\) and \(F_{0}^{B}\) are the set of initial spreaders in layers A and B, respectively.
According to our proposed model, a node can become afflicted only if the proportion of its afflicted intraneighbors and/or interparents exceed a given threshold. Therefore, not all the binary vectors of N_{A}+N_{B} elements represent a feasible state of the process. Note that a node of layer A may become afflicted due to the consequences of the initial spreaders in layer A, layer B, or both. The goal of this paper is to analyze the impact of concurrent phenomena propagation in MIDNs to gain a better understanding of the most robust interdependent network topology. Our propagation model and the Markov model provide a general framework that allows for further investigation of different types of phenomena propagation/impact.
Transition probabilities
Based on our previous definitions, we calculate the onestep transition probability matrix of the process \(\phantom {\dot {i}\!}\wp \in \{0,1\}^{S_{T}\times S_{T}}\), whose generic element \(\wp _{s,s^{\prime }}\) gives the probability that the network transits from state s to state s^{′}. We let \(\Delta s=s^{\prime }s\). The jth element of vectors Δs and s are represented by Δs_{j} and s_{j}, respectively. In order to calculate \(\wp _{s,s^{\prime }}\), we need to identify two types of transitions: i) transitions where Δs_{j}=0, and ii) transitions where Δs_{j}=1. We denote an indicator function I(cond) where cond is a boolean value and I(true)=1, I(false)=0. A formal definition for \(\wp _{s,s^{\prime }}\) is provided in Eq. 3,
where f(j) denotes the probability that there is no change in the jth component of the network state when transitioning from state s to state s^{′}, and f^{′}(j) is the probability that there is a change of status for the jth node of the network, going from a working node to a afflicted node. It is important to note that Δs_{j}=0 happens in two cases: i) the related node is already afflicted, as we do not consider recovery or restoration of the network in this paper, and ii) the related node is currently working and it will remain the same as the proportion of afflicted neighbors/parents does not meet the threshold or, although the propagation threshold is met, the propagation did not occur in the current timestep (referring to the probabilistic nature of the propagation model).
We also calculate the value of f(j) and f^{′}(j) by separating the terms related to layers A and B. In particular, the probability of having a change in the jth element of the interdependent network when the network state changes from s to s^{′} can be calculated in the definition for Eq. 4.
Then, g_{a}(j) can be calculated as defined in Eq. 5,
where \(\bar {\pi }(j) = 1  \pi (j)\) and \(\bar {\pi }^{\prime }(j) = 1  \pi ^{\prime }(j)\), with π(j) and π^{′}(j) defined in Eqs. 1 and 2, respectively.
The term g_{b}(j), will be similarly calculated for the nodes located in layer B, by replacing k_{aa} with k_{bb}, and k_{ba} with k_{ab} in Eq. 5. Note that according to our definition, the nodes of the MIDN system are enumerated from 1 to N_{A}+N_{B}, therefore, k_{aa} and k_{ba} are only defined for j≤N_{A} and k_{bb}, k_{ab} are only defined for N_{A}<j≤N_{A}+N_{B}.
The term f^{′}(j) denotes the probability that there is a change in the jth component of the state vector, when the network is transitioning from state s to state s^{′}. We split f^{′}(j) in the contributions related to the two layers A and B. Therefore,
Similar to our previous note, we can calculate the term \(g^{\prime }_{a}(j)\) as follows,
Using a similar argument, we can write similar equations for the nodes in layer B, i.e. \(g^{\prime }_{b}(j) \) can be calculated.
Note: Based on the definition of transition probabilities, we observe that the speed of phenomena propagation will increase, if and only if f^{′}(j)≫f(j) in Eq. 3. This is because, under this condition, the second term in Eq. 3 will dominate the transition probability, meaning that the probability that Δs_{j}=1 is much larger than that of Δs_{j}=0, therefore the number of afflicted nodes increases faster over time. Looking at f(j) and f^{′}(j), we can see that both are functions of p_{max(∗∗)}. By increasing the value of p_{max(∗∗)}, we will have f^{′}(j)>f(j), as f^{′}(j) is an increasing function of π and π^{′}, and thus an increasing function of p_{max(∗∗)}. On the other hand, f(j) is a decreasing function of these probabilities.
Expected absorption time
Phenomena propagation in an MIDN system can be seen as an absorbing Markov chain, where the absorbing states are defined as one of the following: i) all nodes of the MIDN system are afflicted, and ii) no working node can meet the phenomena propagation condition. The set of absorbing states depends on the sets of initial spreaders \(\left (F_{0}^{A}\ \text {and}\ F_{0}^{B}\right)\) and the propagation thresholds. Therefore, we may get different absorbing states for the same network topology when the set of initial spreaders in A and B are different. Given that we have the states of the network and transition probabilities, we can use standard techniques, used in (Charles et al. 1997), for analyzing our proposed Markov process and to calculate the expected absorption time. However, the state space for a reasonably sized network would be very large as we are considering the state of a network’s individual elements. Due to this limitation, we have only tested our model using networks of a relatively small size for the proposed Markov model. We then use this to validate the correctness of our simulations that have been built to handle networks of larger sizes while using the same phenomena propagation model. An example of simulation validation can be found in (Khamfroush et al. 2016). As the focus of this work is to use extensive simulations to provide useful guidelines on the vulnerability of different network topologies, we focus on simulation setup and results in the following sections.
Description of experiments
We investigate how different initial spreader selection strategies affect phenomena propagation in MIDN systems. To study this, we perform and analyze extensive simulations. To be thorough, we consider the time evolution of phenomena propagation of MIDN systems with varying network topologies, coupling strategies, interedge density, average degree, and the choice of initial spreaders. For this work, we are particularly interested in how singlelayer initial spread in Scenarios 11 and 21 compares to multilayer initial spread in Scenarios 12 and 22.
Tested scenarios
For this work, we consider an MIDN topology composed of two layers A and B. For our simulations, we considered four possible scenarios for initial spreader selections. In each scenario considered, we designate F_{0}=⌊Φ·(N_{A}+N_{B})⌋ where Φ is the percentage of nodes we select to be initial spreaders. These scenarios incorporate what we call “singlelayer selection” or “twolayer selection”. If a scenario uses the former, then layer A will have F_{0} nodes selected as initial spreaders; if a scenario uses the latter, then layers A and B will both have F_{0}/2 nodes selected as initial spreaders. Below are the scenarios considered:

Scenario 11: Singlelayer selection, initial spreaders chosen at random.

Scenario 12: Twolayer selection, initial spreaders chosen at random.

Scenario 21: Singlelayer selection, initial spreaders chosen by high degree centrality.

Scenario 22: Twolayer selection, initial spreaders chosen by high degree centrality.
Interconnectivity models
For our simulations, we consider a variety of interconnectivity for thorough examination of our core question. With this in mind, we consider the notion of interconnections (represented by directed interedges) being established at random or by design. Additionally, we also aim to explore how low or high numbers of interconnections affect phenomena propagation.
There are a lot of considerations in our interconnectivity model. First, we introduce the four interconnectivity cases considered in our experiments:

Sparse & Random: In this model, 8% of nodes are selected to have interedges directed to nodes in the other network. 4% of nodes in layer A (chosen at random) have interedges to nodes in layer B (chosen at random), and vice versa.

Dense & Random: In this model, 20% of nodes are selected to have interedges directed to nodes in the other network. 10% of nodes in layer A (chosen at random) have interedges to nodes in layer B (chosen at random), and vice versa.

Sparse & Designed: In this model, 8% of nodes are selected to have interedges directed to nodes in the other network. 4% of nodes in layer A (based on degree centrality) have interedges to nodes in layer B (based on degree centrality), and vice versa.

Dense & Designed: In this model, 20% of nodes are selected to have interedges directed to nodes in the other network. 10% of nodes in layer A (based on degree centrality) have interedges to nodes in layer B (based on degree centrality), and vice versa.
Additionally, in the case of designed interconnectivity (SparseDesigned or DenseDesigned), we consider how nodes are chosen to have interedges. We are interested in exploring how nodes of high degree and low degree affect the evolution of phenomena propagation. For this, we consider three cases of how designed selection of nodes with interedges takes place:

MaxMax: Nodes of highest degree in the currently considered network are designed to have interedges to nodes of the highest degree in the other network, and vice versa.

MaxMin: Nodes of highest degree in layer A are designed to have interedges to nodes of the lowest degree in layer B and nodes of lowest degree in layer B have interedges to nodes of highest degree in layer A.

MinMin: Nodes of lowest degree in the currently considered network are designed to have interedges to nodes of the lowest degree in the other network, and vice versa.
Tested network topologies
Synthetic topologies
For synthetic topologies, we use three wellstudied random generative models for network topologies: ErdösRényi (ER) model (Erdos and Rényi 1960), BarabásiAlbert (BA) model (Barabási and Albert 1999), and the WattsStrogatz (WS) model (Watts and Strogatz 1998). The ER model produces what are commonly referred to as purely random graphs; the BA model produces topologies that maintain a power law degree distribution; and the WS model produces topologies with the smallworld property.
To be thorough, we consider all permutations of the random generative models for a twolayer MIDN topology. For instance, an MIDN topology composed of two layers A and B that are both constructed by the ER model will be referred to as an ERER topology (coinciding with AB). It is important to note that network topologies are not transitive in our experiments — i.e., an ERSW topology is not equivalent to an SWER topology. The reason for this is that a singlelayer selection scenario only selects initial spreaders in layer A, thus a simulation using an ERSW topology is not comparable to a simulation using an SWER topology. To be clear, in all of our experiments, we generate a synthetic, random MIDN topology in each MonteCarlo run.
Realworld topologies
For comparison, we also run simulations using realworld multiplex topologies. These topologies come from different domains — social networks, genetic networks, etc. A key difference between these topologies and our synthetic topologies is the interconnection between layers A and B. In our synthetic topologies, we consider Sparse/Dense and Designed/Random interconnections. In our realworld topologies, each node exists across all layers with interedges to each other node corresponding to it across all layers. For a concise description o fhte realworld topologies considered for this work, refer to Table 2.
Simulation setup
Here, we review some of the constants that are maintained across all simulations considered for our experiments and review some other experimental choices that have not been addressed thus far. For any simulation setup, we consider the evolution of phenomena propagation over a time horizon of T=200 time steps. In the plots presented for our results, we let the xaxis represent the time steps comprising the time horizon T; while the yaxis represents the number (or percentage) of afflicted nodes in the entire MIDN topology.
Additionally, for our synthetic networks, we consider cases where N_{A}=N_{B}=100 and N_{A}=N_{B}=500. For all synthetic topologies, we consider an average degree 〈k〉=4. We evaluated how average degree affects phenomena propagation before making this decision. For our considered threshold parameter values, we observe that higher average degree corresponds with significantly less affliction throughout the phenomena propagation process — refer to Fig. 1.
Additionally, we are interested in the cases where values for our probabilistic thresholdbased model are i) the same for layer A and B and ii) where they are different. The former case is what we refer to as homogeneous propagation while the latter case is referred to as heterogeneous propagation. For heterogeneous propagation, we use the following threshold values: k_{aa}=0.5, k_{bb}=0.2, k_{ab}=0.3, and k_{ba}=0.8. For homogeneous propagation, we use the following threshold values: k_{aa}=k_{bb}=0.3 and k_{ab}=k_{ba}=0.5. These threshold values are used for all simulations, both synthetic and realworld.
Results
To reiterate, we consider a variety of parameters for a simulation. To cope for randomness, each simulation setup is run a number of M=100 Monte Carlo runs, with results being averaged across these runs. Due to the sheer number of parameters considered, our results span hundreds of simulation setups. Therefore, we want to highlight important pieces of data analysis in hopes that key conclusions can be made. So, we elect to show plots of some of the more interesting simulation results and will describe general trends throughout. For brevity, we say “singlelayer selection” to refer to Scenarios 11 and 21 and “twolayer selection” to refer to Scenarios 12 and 22.
Interestingly, for the interconnectivity cases we consider for this work, a noteworthy trend across most synthetic simulation setups is that twolayer selection of initial spreaders appears generally to be more effective than singlelayer selection in affecting the entire MIDN topology. In many cases, this selection strategy afflicts the entire MIDN topology early in the propagation process; whereas the propagation under singlelayer selection commonly saturates early in the propagation process without afflicting the entire topology (or full affliction is attained after twolayer selection). As an additional observation, we observed that as we experimentally increased average degree 〈k〉, singlelayer selection outperforms twolayer selection — this was observed before designating 〈k〉=4 but we felt worth mentioning. We also note that as Φ increases, generally twolayer selection outperforms singlelayer seleciton.
Homogeneous simulations
Here we investigate the results of our homogeneous simulations. As a reminder, we refer to homogeneous simulations as simulations where the threshold values for our propagation model are constant across both layers A and B in the layers of our MIDN topology.
Intuitively, Sparse interconnectivity essentially dampens the possible effects of phenomena propagation under our model. Our results reflect this intuition. Refer to Fig. 2 for a sample of these simulation results. For a detailed overview of affliction w.r.t. to time evolution, refer to Table 3. Overall, we see in the homogeneous case, that Scenario 22 (twolayer selection by degree centrality) is the most effective (w.r.t. to time and percentage of affliction) at reaching near full affliction across most interconnectivity cases considered.
Designed maxmax
We observe for homogeneous simulations with Dense interconnectivity that the variance across MonteCarlo runs is incredibly small across all considered topologies, network sizes, and Φ. This observation makes sense upon further consideration. Consider the BA random model for network topologies. Due to preferential attachment, nodes are more likely to coalesce and associate with nodes of high degree (forming “hubs”) — which are selected to exhibit interconnection under Scenarios 21 and 22. If the central, highly connected node of this hub is afflicted, its lesser connected neighbors are more vulnerable to affliction because that high degree node likely makes up a sizable proportion of its neighbors (thus having more impact on its threshold to affliction). For homogeneous simulations with Sparse interconnectivity, we also see very little variance — though not as small.
For Dense interconnections, generally the singlelayer selection of initial spreaders outperforms the twolayer selection strategy. The difference between these two strategies in this setup is not particularly notable, but the general trend is that singlelayer selection is slightly more effective. Meanwhile, for Sparse interconnections, the general trend holds. However, the difference between singlelayer and twolayer selection strategy is more prominent. Additionally, it is important to note that as Φ increases, the twolayer selection strategy begins to outperform the singlelayer selection strategy. This implies that if you have more budget to select more initial spreaders under these parameters, twolayer selection would be most effective. However, if you have a more conservative budget, singlelayer selection would be more appropriate.
Designed maxmin
For both Dense and Sparse interconnectivity, we observe that our averaged data shows that twolayer selection notably outperforms singlelayer selection in all cases. Under Dense interconnectivity, the variance of the resulting phenomena propagation for singlelayer selection is notably large; while the variance of the resulting phenomena propagation for twolayer selection in this case is very small. However, under Sparse interconnectivity, the variance of the resulting phenomena propagation for singlelayer selection is significantly smaller than under Dense interconnectivity.
It is important to note, that under this interconnectivity, twolayer selection results in fullaffliction in nearly every case. Singlelayer selection is not as effective, with no simulation using singlelayer selection having comparable affliction rates to twolayer selection. The performance of singlelayer selection is starkly less impressive under Sparse interconnectivity — with propagation saturating early on with about 50% affliction across all cases. This is likely due to initial spreaders exclusively selected in layer A propagate affliction to nodes minimum degree nodes in layer B. As a result, layer A is roughly 90% afflicted whereas layer B roughly 10% afflicted.
Designed minmin
With an increase in the size of the networks, singlelayer selection is notably more effective than in smaller network sizes — reaching full affliction (on average) in some cases. Otherwise, results for this set of experiments is very comparable to Designed MaxMin. However, there does appear to be less variance for singlelayer selection. In the MaxMin case, some nodes in the layer B can crosspropagates though maximum degree nodes under twolayer selection. However, in the case of MinMin, it this does not happen. This is likely why the difference is not as initially prominent.
Random
There is a stark difference between Dense interconnectivity and Sparse interconnectivity in this case. In Dense interconnectivity, the performance between singlelayer selection and twolayer selection is marginal — though twolayer selection has a slight edge. Both selection strategies, on average, result in full affliction (or very close to it) in all cases considered — when N_{A}=N_{B}=500 all experiments reached full affliction. Additionally, variance of performance was very low under Dense interconnectivity. It should be noted that when Φ is a low value (e.g., Φ=3%), there is a split in certain topologies where singlelayer selection performs better in terms of overall affliction. However, as Φ increases, twolayer selection begins to close the gap in these topologies and eventually overtakes it in terms of affliction w.r.t. to time.
These remarks cannot be made for this experiment under Sparse interconnectivity. When N_{A}=N_{B}=100, the performance of singlelayer selection would typically hover in the range of about 50%70% affliction.
Heterogeneous simulations
Here we investigate the results of our heterogeneous simulations. As a reminder, we refer to heterogeneous simulations as simulations where the threshold values for our propagation model are unique across both layers A and B in the layers of our MIDN topology. There are no standout trends across the different models of interconnectivity. Below is a brief summary of the general behavior of these simulations. Generally, for the Heterogeneous case (under the threshold values we use), the most apparent trend is that singlelayer selection seems to be a more potent selection strategy overall than twolayer selection. It is important to note that this trend is not guaranteed to hold for different threshold values under this propagation model.
For this work, under singlelayer selection, initial spreader selection takes place in layer A, which is more robust to affliction by interparents and intraparents. Since the selection is concentrated in the more robust layer, the results show that we get more affliction in that layer. Subsequently, the crosspropagation is not as difficult since the layer A is more vulnerable. As for twolayer selection, less affliction occurs in layer A due to its robustness since half of the resources for selection are used on the vulnerable layer. For a more detailed overview of affliction w.r.t. to time evolution, refer to Table 4.
For both Sparse and Dense interconnectivity, when N_{A}=N_{B}=100, the general trend is that twolayer selection outperforms singlelayer selection when the percentage of nodes selected to be initial spreaders is smaller. Once 10% of the nodes are selected to be initial spreaders, for both cases, the singlelayer selection strategy begins to overtake the twolayer selection strategy. However, the difference in performance is marginal and not highly significant.
However, when N_{A}=N_{B}=500, under Dense interconnectivity, the strategies perform nearly identically until 10% of nodes are selected as initial spreaders. In which case, singlelayer selection begins to have the advantage. The advantage in this case, is notable and is quite significant when compared to the experiments where N_{A}=N_{B}=100. Also, the variance in the resulting phenomena propagation is very high when the percentage of nodes selected to be initial spreaders is small.
Realworld simulations
Results for realworld simulations depart from the general trend we observe in our synthetic simulations. In our realworld simulations, under designed initial spreader selection (Scenario 2), we observe that singlelayer initial spreader generally outperforms multilayer initial spreader selection. However, the important detail to note is that the difference is not as notable as with our synthetic simulations. Additionally, the propagation process converges in a few time steps over the entire time horizon T. The amount of affliction that occurs varies based on value of Φ. In Scenario 22, typically the affliction struggles to afflict additional nodes beyond the set of initial spreaders. However, in the other Scenarios considered, we see that affliction typically saturates at around 2·F_{0}. The time evolution of the propagation process for these realworld topologies can be seen in Fig. 3.
The reasoning for this is likely due to the interconnection of the realworld topologies. It is very difficult to find publicly available MIDN topologies. ComuneLab provides publicly available topological data for multiplex networks, from which we acquired realworld MIDN topological data. However, a key distinction between the realworld topologies we run simulations on from this resource is that they are multiplex networks. Refer to Table 2 for an overview of the realworld multiplex topologies considered for this work.
Multiplex networks are topologies with numerous layers, with each nodes existing across every layer. These multilayer network models allow for interesting relationships between entities. For instance, a layer in a multiplex network can have edges that represent who retweets who on Twitter, while another layer in the same multiplex network can have edges that represent who replies to who on Twitter. In this example, each node is shared across each layer and interedges are only between each node to itself in every other layer. For instance, Donald Trump’s Twitter account would be a node shared across all layers in a multiplex Twitter network. The interedges this node has will be to the other nodes representing Donald Trump’s Twitter account across all other layers. These nodes will have no interedges to nodes representing Twitter user accounts other than Donald Trump’s.
This detail is in stark contrast to our synthetic networks. We vary the level of interdependency across our synthetic topology through Sparse/Dense and Random/Designed interconnections. A visualization of what these topologies look like can be found in Fig. 4. Due to this high interconnection between layers, it makes sense that these networks do not exhibit the same behaviors of affliction as our synthetic topologies.
Degree singlelayer vs. random multilayer
For this work, we are interested in how designed, singlelayer selection and random, twolayer selection compare. The interest in this is motivated by the notion that it is theoretically easier to obtain topological structure of one layer within an MIDN topology. With this knowledge, one could target highly connected nodes in this layer for affliction. With this in mind, we want to compare how a singlelayer selection of initial spreaders by degree centrality (Scenario 12) compares to a twolayer selection of initial spreaders chosen at random (Scenario 21).
From analyzing the results from Tables 4 and 3, we see that singlelayer selection by degree centrality (Scenario 21) generally outperforms twolayer selection at random (Scenario 12) in both the heterogeneous and the homogeneous cases in the vast majority of experiments considered. However, it should be reiterated that singlelayer selection selects nodes to be initial spreaders in layer A. With this in mind, k_{ab}=0.3 (and k_{ba}=0.8) for heterogeneous cases. The results of this analysis for the heterogeneous simulations depend largely on the interthresholds. If the interthreshold values are sufficiently large, then Scenario 12 would very likely outperform Scenario 21.
Conclusions
In closing, we studied how initial spreader selection strategies compare by analyzing the evolution of phenomena propagation over time using a stateoftheart thresholdbased propagation model. The benefit of our propagation model is that it is general enough to encapsulate other models of propagation (such as epidemic, linear probabilistic, linear threshold, etc.). Our results show that, if you have knowledge about the topological structure (such as degree of nodes), multilayer designed selection outperforms singlelayer designed selection in most cases considered by our experiments. However, if you are in a situation where topological information is unavailable, random selection in a singlelayer is more effective than multilayer random selection. Additionally, from our work, we observe that singlelayer selection by degree centrality (Scenario 21) generally outperforms twolayer selection at random (Scenario 12) most cases considered for this work.
Prior works investigating initial spreaders (or seed selection) in multilayer networks are often interested in how phenomena propagation can be maximized or minimized given the context of the problem at hand (such as the influence maximization problem). In this work, our results are focused on analyzing how different interconnectivity cases and selection strategies with respect to layers considered for selection affect phenomena propagation. The general nature of our problem allows our work to provide insights to problems that aim to either maximize or minimize the overall affliction of a MIDN topology as a result of phenomena propagation.
For future works in this direction, it is of interest to investigate the effectiveness of selection strategies in MIDN topologies consisting of m layers. With insight from how these varying strategies compare w.r.t. topological structure and interconnectivity, studying affliction in MIDNs of m layers may be easier to investigate.
Availability of data and materials
The data used for this work was synthetically generated using standard generative models. The models used for this work are outlined in full within the body of the paper. References to the random generative models used have been included in this work. However, we do provide the source code used to run experiment simulations for the results of this work in a publicly accessible GitHub repository.^{Footnote 2}
Notes
We do not consider random (or designed) singlelayer spread with phenomena initially spreading in because it is essentially the same scenario, since these are general network components.
Abbreviations
 BA:

BarabásiAlbert model for random graphs following a power law degree distribution
 ER:

ErdősRényi model for random graphs
 ICM:

Independent cascade model
 MIDN:

Multilayer interdependent network
 PLC:

Programmable logic controller
 SCADA:

Supervisory control and data acquisition
 WS:

WattsStrogatz model for random, smallworld graphs
References
Albert, R, Jeong H, Barabási AL (2000) Error and attack tolerance of complex networks. Nature 406(6794):378.
Albright, D, Brannan P, Walrond C (2010) Did Stuxnet Take Out 1,000 Centrifuges at the Natanz Enrichment Plant?. Institute for Science and International Security. https://isisonline.org/isisreports/detail/didstuxnettakeout1000centrifugesatthenatanzenrichmentplant/.
Anderson, R, Fuloria S (2010) Who controls the off switch? In: 2010 First IEEE International Conference on Smart Grid Communications, 96–101.. IEEE, Gaithersburg. https://doi.org/10.1109/SMARTGRID.2010.5622026. https://ieeexplore.ieee.org/abstract/document/5622026.
Barabási, AL, Albert R (1999) Emergence of scaling in random networks. Science 286(5439):509–512.
Brooks, R, Sander S, Deng J, Taiber J (2008) Automotive system security: challenges and stateoftheart In: Proceeding CSIIRW ’08 Proceedings of the 4th annual workshop on Cyber security and information intelligence research: developing strategies to meet the cyber security and information intelligence challenges ahead, 26.. ACM, New York. https://dl.acm.org/citation.cfm?id=1413170.
Charles, M, Grinstead J, Snell L (1997) Introduction to probability. Am Math Soc. https://bookstore.ams.org/iprob/.
Checkoway, S, McCoy D, Kantor B, Anderson D, Shacham H, Savage S, Koscher K, Czeskis A, Roesner F, Kohno T, et al. (2011) Comprehensive experimental analyses of automotive attack surfaces In: Proceeding SEC’11 Proceedings of the 20th USENIX conference on Security, 77–92.. USENIX, San Francisco. https://dl.acm.org/citation.cfm?id=2028073.
Chen, W, Wang Y, Yang S (2009) Efficient influence maximization in social networks In: Proceeding KDD ’09 Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining, 199–208.. ACM, New York. https://dl.acm.org/citation.cfm?id=1557047.
Chen, W, Wang C, Wang Y (2010) Scalable influence maximization for prevalent viral marketing in largescale social networks In: Proceeding KDD ’10 Proceedings of the 16th ACM SIGKDD international conference on Knowledge discovery and data mining, 1029–1038.. ACM, New York. https://dl.acm.org/citation.cfm?id=1835934.
Cohen, R, Erez K, BenAvraham D, Havlin S (2001) Breakdown of the internet under intentional attack. Phys Rev Lett 86(16):3682.
Coleman, J, Katz E, Menzel H (1957) The diffusion of an innovation among physicians. Sociometry 20(4):253–270.
De Domenico, M, Nicosia V, Arenas A, Latora V (2015) Structural reducibility of multilayer networks. Nat Commun 6:6864.
Erdos, P, Rényi A (1960) On the evolution of random graphs. Publ Math Inst Hung Acad Sci 5(1):17–60.
Erlandsson, F, Bródka P, Borg A (2017) Seed selection for information cascade in multilayer networks. arXiv.org 689. https://link.springer.com/chapter/10.1007/9783319721507_35.
Halperin, D, HeydtBenjamin TS, Fu K, Kohno T, Maisel WH (2008) Security and privacy for implantable medical devices. IEEE Pervasive Comput 7(1):30–39. https://ieeexplore.ieee.org/abstract/document/4431854.
Hanna, S, Rolles R, MolinaMarkham A, Poosankam P, Blocki J, Fu K, Song D (2011) Take two software updates and see me in the morning: The case for software security evaluations of medical devices In: Proceeding HealthSec’11 Proceedings of the 2nd USENIX conference on Health security and privacy.. USENIX Association, Berkeley. https://dl.acm.org/citation.cfm?id=2028032.
Hoppe, T, Kiltz S, Dittmann J (2008) Security threats to automotive CAN networks–practical examples and selected shortterm countermeasures In: International Conference on Computer Safety, Reliability, and Security, 235–248.. Springer. https://link.springer.com/chapter/10.1007/9783540876984_21.
Hudson, N, Turner M, Nkansah A, Khamfroush H (2019) On the effectiveness of standard centrality metrics for interdependent networks In: 2019 International Conference on Computing, Networking and Communications (ICNC).. IEEE, Honolulu. https://ieeexplore.ieee.org/abstract/document/8685586.
Khamfroush, H, Bartolini N, La Porta TF, Swami A, Dillman J (2016) On propagation of phenomena in interdependent networks. IEEE Trans Netw Sci Eng 3(4):225–239.
Kandhway, K, Kuri J (2017) Using node centrality and optimal control to maximize information diffusion in social networks. IEEE Trans Syst Man Cybern Syst 47(7):1099–1110.
Kempe, D, Kleinberg J, Tardos É (2005) Influential nodes in a diffusion model for social networks In: International Colloquium on Automata, Languages, and Programming, 1127–1138.. Springer. https://link.springer.com/chapter/10.1007/11523468_91.
Kim, H, Beznosov K, Yoneki E (2014) Finding influential neighbors to maximize information diffusion in twitter In: Proceeding WWW ’14 Companion Proceedings of the 23rd International Conference on World Wide Web, 701–706.. ACM, New York. https://doi.org/10.1145/2567948.2579358. https://dl.acm.org/citation.cfm?id=2579358.
Kim, H, Beznosov K, Yoneki E (2015) A study on the influential neighbors to maximize information diffusion in online social networks. Comput Soc Networks 2(1):3.
Kimura, M, Saito K, Motoda H (2009) Blocking links to minimize contamination spread in a social network. ACM Trans Knowl Discov Data (TKDD) 3(2):9.
McDaniel, P, McLaughlin S (2009) Security and privacy challenges in the smart grid. IEEE Secur Priv 7(3):75–77. https://ieeexplore.ieee.org/abstract/document/5054916.
Metke, AR, Ekl RL (2010) Security technology for smart grid networks. IEEE Trans Smart Grid 1(1):99–107.
Michalski, R, Kajdanowicz T, Bródka P, Kazienko P (2014) Seed selection for spread of influence in social networks: Temporal vs. static approach. New Gener Comput 32(34):213–235.
Mo, Y, Kim THJ, Brancik K, Dickinson D, Lee H, Perrig A, Sinopoli B (2012) Cyber–physical security of a smart grid infrastructure. Proc IEEE 100(1):195–209.
Papanastasiou, Y (2018) Fake news propagation and detection: A sequential model. https://papers.ssrn.com/sol3/Papers.cfm?abstract_id=3028354. Available at SSRN: https://ssrn.com/abstract=3028354 or https://doi.org/10.2139/ssrn.3028354.
PastorSatorras, R, Vespignani A (2001) Epidemic spreading in scalefree networks. Phys Rev Lett 86(14):3200.
Rahman, MA, Bera P, AlShaer E (2012) Smartanalyzer: A noninvasive security threat analyzer for ami smart grid In: 2012 Proceedings IEEE INFOCOM, 2255–2263.. IEEE, Orlando. https://ieeexplore.ieee.org/abstract/document/6195611. https://doi.org/10.1109/INFCOM.2012.6195611.
Rushanan, M, Rubin AD, Kune DF, Swanson CM (2014) Sok: Security and privacy in implantable medical devices and body area networks In: 2014 IEEE Symposium on Security and Privacy, 524–539.. IEEE, San Jose. https://ieeexplore.ieee.org/abstract/document/6956585. https://doi.org/10.1109/SP.2014.40.
Salehi, M, Sharma R, Marzolla M, Magnani M, Siyari P, Montesi D (2015) Spreading processes in multilayer networks. Netw Sci Eng IEEE Trans 2(2):65–83.
Shu, K, Bernard HR, Liu H (2019) Studying fake news via network analysis: detection and mitigation In: Emerging Research Challenges and Opportunities in Computational Social Network Analysis and Mining, 43–65.. Springer. https://link.springer.com/chapter/10.1007/9783319941059_3.
Stark, C, Breitkreutz BJ, Reguly T, Boucher L, Breitkreutz A, Tyers M (2006) Biogrid: a general repository for interaction datasets. Nucleic Acids Res 34(suppl_1):535–539.
Watts, DJ, Strogatz SH (1998) Collective dynamics of ’smallworld’networks. Nature 393(6684):440.
Zhao, D, Li L, Li S, Huo Y, Yang Y (2014) Identifying influential spreaders in interconnected networks. Phys Scr 89(1). https://iopscience.iop.org/article/10.1088/00318949/89/01/015203/meta.
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HK designed and formulated the problem, and proposed approaches to solve the problem described in this work. NH wrote the final draft of the paper for submission and analyzed results of experiments. SI ran simulations, documented results, and wrote an initial draft for the paper. MN collaborated with HK on formulation and simulation design and identifying the credibility of the problem formulation and simulations. All authors read and approved the final manuscript.
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Khamfroush, H., Hudson, N., Iloo, S. et al. Influence spread in twolayer interdependent networks: designed singlelayer or random twolayer initial spreaders?. Appl Netw Sci 4, 40 (2019). https://doi.org/10.1007/s4110901901503
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DOI: https://doi.org/10.1007/s4110901901503