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Analysis and control of information diffusion dictated by user interest in generalized networks
 Eleni Stai^{1}Email authorView ORCID ID profile,
 Vasileios Karyotis^{1} and
 Symeon Papavassiliou^{1}
 Received: 1 June 2015
 Accepted: 4 November 2015
 Published: 2 December 2015
Abstract
The diffusion of useful information in generalized networks, such as those consisting of wireless physical substrates and social network overlays is very important for theoretical and practical applications. Contrary to previous works, we focus on the impact of user interest and its features (e.g., interest periodicity) on the dynamics and control of diffusion of useful information within such complex wirelesssocial systems. By considering the impact of temporal and topical variations of users interests, e.g., seasonal periodicity of interest in summer vacation advertisements which spread more effectively during Spring–Summer months, we develop an epidemicbased mathematical framework for modeling and analyzing such information dissemination processes and use three indicative operational scenarios to demonstrate the solutions and results that can be obtained by the corresponding differential equationbased formalism. We then develop an optimal control framework subject to the above information diffusion modeling that allows controlling the tradeoff between information propagation efficiency and the associated cost, by considering and leveraging on the impact that user interests have on the diffusion processes. By analysis and extensive simulations, significant outcomes are obtained on the impact of each network layer and the associated interest parameters on the dynamics of useful information diffusion. Furthermore, several behavioral properties of the optimal control of the useful information diffusion with respect to the number of infected/informed nodes and the evolving user interest are shown through analysis and verified via simulations. Specifically, a key finding is that low interestrelated diffusion can be aided by utilizing proper optimal controls. Our work in this paper paves the way towards this usercentered information diffusion framework.
Keywords
 User interests
 Information diffusion
 Generalized networks
 SIS epidemic model
 Timevarying interests
 Optimal control
 Pontryagin’s Maximum Principle
 Hamilton–Jacobi–Bellman equation
Introduction
Analyzing and controlling information diffusion in complex networks is of high research and practical interest nowadays. “Information” may appear in diverse forms, useful or malicious, each with different diffusion dynamics and demanding different types of control. Malicious information, e.g., a dangerous computer virus, might have catastrophic outcomes calling for suppressive control, while marketing advertisements can be exploited for maximizing online revenues and may be enhanced by an amplifying type of control.
To better facilitate the increased needs for effective information exchange, continuing technological advances in wireless and wired communications and the development of online social networks have given rise to “generalized” network systems. The latter consist of a physical layer, i.e., a wireless medium, and a social overlay, where social encounters develop, forming combined cyberphysical, e.g., socialwireless networks, referred to as generalized networks [1]. According to [2], generalized networks, even when consisting of a physical (e.g., wireless multihop) and only one social network can significantly improve information spread. In this paper, we focus on socialwireless types of generalized networks, while other types may be straightforwardly considered.
Motivated by the above observations on networks and information proliferation, in this paper, we focus on the diffusion dynamics and control of useful information in generalized networks. Various relevant works on the topic exist in the literature ("Related work and contributions"). However, albeit, they bear a specific drawback by not accounting for the evolution of user interest on the information diffused. Typically, humans interact with each other and exchange content on the basis of features such as “topics of information”. In particular, during an encounter, humans may not care for information that is out of their interest range at that particular time, thus not participating in the diffusion of the corresponding topic. Therefore, communicating information is highly affected by user interests and their temporal evolution, since not every contact does necessarily imply information transfer for all the topics under diffusion. It rather depends both on human preferences and their interconnections (physical and social topology).
Several realworld examples indicate the dependence of information diffusion dynamics on the temporal and topical variation of user interests [3–5]. For instance, advertisements on summer vacations are expected to have a more successful spread outcome during the Spring and Summer months, while being hobbled during Fall and Winter months, highlighting an emergent seasonal periodicity with respect to user interests. Secondly, news on a soccer match might not be well spread within the members of a dance group, while they are expected to be quickly spread within the members of a soccer club. The first of the above cannot be expressed by the current models of information diffusion which do not segregate the diffusion success rate with respect to seasonal dependence, while the second case implies a nonhomogeneous information rate across populations with different characteristics. As a result, for a realistic inclusion of users’ interests in the information diffusion model, the interests should be considered time varying, e.g., reflecting the evolving seasonal behavior of human beings [6]. The second example further implies the need of explicitly taking into account the subject of users’ interests, when designing information diffusion models.
Thus, in this paper we introduce and develop for the first time an information diffusion modeling framework that takes into account both user interests’ differentiation and their possible temporal variability (e.g., periodicity). Furthermore, we provide an optimal control framework on top of the information diffusion model that allows for tradingoff diffusion efficiency with the associated cost, leveraging on the impact that user interest has on information dissemination dynamics. To the best of our knowledge, there is limited literature in the field of optimal control over diffusion dynamics described by epidemic modeling with timeevolving parameters [6, 7]. Incorporating control, will benefit information spreading, particularly when there is limited interest on the useful information being diffused, in which case, it can be mapped to, e.g., advertising campaigns or other incentives provided to users in an optimized way with respect to cost. An example of explicit control is the provision of incentives to users, e.g., in the form of competition, rewards, reputation, etc., to participate in information propagation when their interest itself in the propagated topic is limited, decreasing in this way the probability that information propagation on a specific subject deceases fast enough. Significant outcomes are provided on the impact of each topological layer (social or wireless) and the associated interest parameters on the dynamics and control of information diffusion over complex socialwireless topologies, via analysis, numerical evaluations and simulations of relevant scenarios. Furthermore, the properties/behavior of the optimal controls on information diffusion are extensively studied.
The rest of the paper is organized as follows. "Related work and contributions" describes related literature and positions our work within the existing relevant literature, while "System model, notation and assumptions" presents the employed system model. "Information diffusion modeling and analysis without control" analyzes the proposed information diffusion model and the examined application scenarios. In the sequel, "Optimal control framework for information diffusion" introduces the information diffusion optimal control framework, while "Simulation and numerical results without applying control" and "Simulation results, numerical results and discussion in controlled information diffusion" present and thoroughly discuss the performed simulation results and numerical evaluations without and with control, respectively. Finally, "Conclusions" concludes the paper.
Related work and contributions
Due to the importance of information nowadays, studying the properties of its diffusion along with the possibility of control has attracted considerable interest. In this paper, we focus on two important facets of information diffusion, namely the dynamics of information spreading and its optimal control.
Regarding the dynamics of information diffusion, the earliest and most frequently encountered approaches were inspired by epidemiological models [8, 9]. Some of the most recent ones are [1, 10–13], while more can be found in the references therein. More specifically, both stochastic and deterministic epidemic models exist for information propagation [7, 14], where the nodes having received the information are denoted as “infected”. Stochastic epidemic models treat information propagation as a discrete time process (Discrete Time Markov Chain) [7, 14] being more suitable for smallscale systems whereas deterministic epidemic models assume continuous processes relying on the law of large numbers and applying differential equations or inclusions [15, 16], thus being more suitable for largescale systems. In this paper, we will apply a deterministic epidemic model. Most of deterministic models consider the evolution of the cumulative system state/number of infected individuals (macroscopic modeling), denoted as “population dynamics”, assuming homogeneous infection rates for all population members. On the other hand, the deterministic “network” models study the state of each individual separately and also segregate infection rates between different pairs of individuals [7]. However, system state transitions (i.e., population dynamics) depend on the state transition models developed for each individual, e.g., susceptible–infected–susceptible (SIS), etc. [7, 8]. Contrary to deterministic network models, a typical assumption when considering population dynamics is that of homogeneous mixing, where contact patterns between individuals are considered highly homogeneous [17]. Both types of models, stochastic and deterministic, account for the endogenous (transition that takes place owe to internal individual operation, e.g., recovery transition) and exogenous (transitions dictated by external factors, e.g., infection transition) transition rates expressing the topological and operational, endogenous or exogenous, factors that affect the evolution of the system [7].
Information dissemination epidemic models have been developed for different network topologies, e.g., wireless networks [18], social networks [8] and multiple social networks [3], and generalized networks [1]. More specifically, epidemic models, e.g., SIS, susceptible–infected–removed (SIR) and susceptible–infected (SI) [6, 8] have been adopted and adapted over diverse network topologies to describe the spreading of useful or malicious information. In this work, we mainly focus on the diffusion of useful information over generalized networks based on the SIS epidemic model. Our model lies between the frameworks of population dynamics and network models, since we study system state transitions while considering neighborhood relations in a nodedegree sense.
Furthermore, to the best of our knowledge, the impact of user interests and their temporal variability analyzed in this work have been considered in the literature in a limited degree, e.g., [3, 4, 19]. In [3], the authors aim at finding the minimum number of seed users who can spread the information to all users interested in the specific topic over multiple online social networks (where some users belong in more than one online social networks simultaneously). The work in [19] studies the role of information diffusion to the evolution of the network topology considering the link formation process with respect to sources/retransmitters of the information, based on users’ preferences. User interests in the information topics being propagated are also inferred and considered in [4] to detect active links in the diffusion of a given message over the network. Generally, most of the previous works, except from taking into account contactrelated and topological factors affecting information diffusion dynamics [1, 13], they occasionally regard static users’ interests [3, 4], but not the user interest temporal variability. In a closer spirit to our approach, [6] studies the spatiotemporal dynamics of information diffusion via partial differential equations, while incorporating time decreasing users’ interest on the propagated messages. A similar study is performed for the case of malware dissemination over the Internet in [15]. Specifically, in [15], the infection rate decreases with time while this time dependence is shown via experimentation to model better the Code Red worm propagation. The reason for such decreasing infection rate is that worm spread over the Internet can be slowed down by countermeasures employed by users and congestion points arising over Internet. As in [15], in our approach, the introduction of the time evolving users’ interests in the information propagation decisions along with the consideration of a network substrate abolishes the homogeneity assumption. However, contrary to [15] and [6], in this work we are not restricted to decreasing with time users’ interests, but we apply diverse function forms of the latter (e.g., periodic).
Apart from analyzing the spreading, controlling the information diffusion over various types of networks via explicit, e.g., [7, 10, 20–22] or implicit control, e.g., [16], is highly important. A thorough overview of the current framework of controlling epidemics can be found in [7], including heuristic feedback methods and optimal control policies for both population dynamics and deterministic network models along with spectral control policies for the latter. The authors usually adopt optimal control frameworks for obtaining features allowing the control of the corresponding diffusion properties, which are modeled via differential equations (deterministic models). The work in [20], studies the possible attack strategies of malware over wireless networks and the extent of damage they can sustain. The control parameters consist of the transmission range and media scanning rate of the worm targeting at accelerating its spread. Malware information dissemination is also studied in [10], where the control signal distribution time is determined, aiming to minimize the number of infected nodes and the cost of control. Similar approaches for malware quarantining and filtering (e.g., configurable firewall) are developed and analyzed in [21–23]. These problem approaches, although different in various scopes compared to the target of this paper, they resemble and serve as driving forces to our proposed model and analysis.
Considering implicit control, in [16], malware information propagation is studied and analyzed over a homogeneous mixing network, where control takes the form of updates to nodes from an external source to which nodes reply via a best response (game theoretic) scheme. Also, in [16], there is an implicit introduction (i.e., via the timevarying state of the system) of timevarying behavior on the parameters of the information diffusion epidemic model. However, this takes place in a more restricted sense and with a different scope (i.e., malware propagation) compared to our work.
Our work dealing with nonmalicious information, identifies a major driving force for the successfulness of diffusion, namely user interest and its temporal properties, opening up new directions for the optimal control of useful information diffusion taking into account these aspects as well. Moreover, although our approach adopts a similar problem formulation and analytical approach as in [20], contrary to [7, 10, 20–22], the state constraint, i.e., the epidemicbased differential equation of the evolution of the number of infected nodes, has timevarying parameters due to the temporal dependence of users’ interests considered in this paper.
System model, notation and assumptions
We consider a wireless multihop network of N nodes uniformly and independently distributed on a square region of side L. Each node has a transmission radius R. For simplicity, mobility is ignored, since compared to an MMS type of information spreading, the corresponding longrange information spreading achieved by mobility, which is essentially of P type, will lead to a similar, but smaller effect, as also argued in [1].
We assume M classes of information denoted by \(m=1...M\), as in [24]. Each class consists of messages on a specific topic, e.g., summer vacations advertisements. Information diffusion is studied separately for each class; however, interactions among separate classes are taken into consideration in the information diffusion model’s probabilistic setting. Each node i is characterized by its interest in class m at time t, denoted as \(R_i^m(t)\), where \(\sum _{\forall m} R_i^m(t)\le 1, ~\forall i\) (e.g., normalization over all classes). The information diffusion process proposed in this paper is based on the Susceptible–Infected–Susceptible (SIS) epidemic model [9]. We consider the following mapping. A node i is considered Infected (i.e., informed) for a specific class m of information if it possesses at least one message belonging in this class, otherwise i is considered Susceptible (i.e., not informed) for class m. This means that an informed/notinformed node is mapped to an infected/susceptible state correspondingly, in epidemiology terms. More precisely, the transition from the susceptible state to the infected state for a particular class takes place when a node receives information about this class, while an infected node transits back to the susceptible state when it deletes all messages for this class.
Notation and explanation of symbols.
Symbol  Interpretation 

\(I^m(t)\)  Number of Infected nodes for class m 
\(S^m(t)\)  Number of Susceptible nodes concerning class m 
\(\mathcal {I}^m(t)\)  Set of Infected nodes concerning class m 
\(\mathcal {N}_S(i)\)  Set of node i’s friends in the social layer 
\(\mathcal {N}_{P}(i)\)  Set of connections of i in the wireless network (physical layer) 
\(0\le p_1,p_2,q\le 1\)  Probabilities defined in the proposed information diffusion model 
\(N_S^{avg}\)  The average degree of all nodes in the social layer 
\(f_1(x)\)  \(f_1(x):[0,1]\rightarrow [0,1]\) monotonically increasing on x 
\(f_2(x)\)  \(f_2(x):[0,1]\rightarrow [0,1]\) monotonically decreasing on x 
\(f_{3}(x)\)  \(f_{3}(x):[0,1]\rightarrow [0,1]\) monotonically increasing on x 
Information diffusion modeling and analysis without control
 1.For the classes for which i is infected/informed:
 (a)
i diffuses information about class m with probability \(f_1(R_i^m(t))\) (also denoted as \(f_1(t)\) for simplicity),
 (b)
i deletes all messages about class m with probability \(q f_2(R_i^m(t))\), where the parameter q is introduced to control the deletion process and \(f_2(R_i^m(t))\) will be also denoted as \(f_2(t)\) for simplicity.
$$\begin{aligned} \sum _{m:~i\in \mathcal {I}^m(t)} (f_1(R_i^m(t))+q f_2(R_i^m(t)))\le 1. \end{aligned}$$  (a)
 2.
Node i performs another action—which is not of interest for the information diffusion—with probability equal to \(1\sum _{m:~i\in \mathcal {I}^m(t)} (f_1(R_i^m(t))+q f_2(R_i^m(t)))\).

with probability \(p_1\), node i employs an MMS type of transmission, including as receivers each \(j\in \mathcal {N}_S(i)\) selected with probability \(f_{3}(R_j^m(t))\) (also denoted as \(f_3(t)\) for simplicity), where \(f_{3}(R_j^m(t))\) for all \(j\in \mathcal {N}_S(i)\) does not form a probability distribution,

with probability \(p_2\), node i broadcasts to all its \(\mathcal {N}_{P}(i)\) neighbors (Ptype action).
The above diffusion process requires that there is always an infected node for every class to maintain the spreading. However, all infected nodes for a particular class may delete their information for this class, thus disrupting its diffusion. Exogenous impact such as the optimal control which will be introduced in "Optimal control framework for information diffusion", may be leveraged to alleviate in a certain degree such phenomena of extinction of a whole information class. In the case of Ptype contacts, we do not consider the interests of users receiving a Pinduced message, as the latter is broadcasted indiscriminately to all of them.
The ODE (1) has a unique solution when \(f_{1}^{\mathrm{avg}}(t),~f_{2}^{\mathrm{avg}}(t), ~f_{3}^{\mathrm{avg}}(t)\) are continuous functions with respect to time (Cauchy–Lipschitz Theorem [25]). The righthand side is obviously Lipschitz continuous with respect to \(I^m\). This fact has an impact on the design of possible forms for the interests’ functions \(R_i^m(t),~\forall m,i\), which should be continuous in time. It also has impact on the design of possible formats for the functions \(f_{1}^{\mathrm{avg}}(t),~f_{2}^{\mathrm{avg}}(t), ~f_{3}^{\mathrm{avg}}(t)\).
Scenario 1: periodic users’ interests
In this scenario, two classes of information are considered. The time continuous interests’ functions take sinusoidal forms to express users’ time periodicity of their interest with respect to the propagated information. Specifically, \(R_i^1(t)=1A_i\sin ^2(a(t+b_i))+B_i, ~\forall i,\) for class \(m=1\), where \(a>0\) determines the period of users’ interests and \(A_i,~b_i,~B_i\) are appropriately defined constants. Then, \(R_i^2(t)=A_i\sin ^2(a(t+b_i))B_i, ~\forall i\), so that \(R_i^1(t)+R_i^2(t)=1,~\forall t,i\). Note that, we consider the same frequency for all sinusoidal interests assuming the propagation of information that intrigues the attention of all users over specific time periods such as vacations, summer sports, Halloween, etc.
We can also assume that \(f_{1}^{\mathrm{avg}}(t), ~f_{2}^{\mathrm{avg}}(t),~f_{3}^{\mathrm{avg}}(t)\) represent the expected values of the corresponding functions of users’ interests at time t. Thus, users’ interests will vary randomly according to a distribution with mean value \(1A\sin ^2(a(t+b))+B\) for class 1, letting the complementary interest (i.e., with mean value \(A\sin ^2(a(t+b))B)\) to be assigned to class 2.
Scenario 2: comparison of information diffusion dynamics among groups with different characteristics
In this scenario, we apply constant interests to study how information of a specific subject spreads in groups characterized by different features such as in the second example described in the introductory section ("Background"). This special case is similar to the SIS models developed in literature [8, 9] in the sense that the parameters applied in the ODEs describing the dynamics of information diffusion are constant, contrary to the time varying parameters (\(f_{1}^{\mathrm{avg}}(t),~ f_{2}^{\mathrm{avg}}(t),~ f_{3}^{\mathrm{avg}}(t)\)) considered in this paper. Therefore, the already existing schemes [26] constitute special cases of our proposed diffusion model.
In this framework, we consider two groups and one information class (e.g., class 1). For both groups \(R_i^1(t)=a, ~\forall i\), \(0<a<1\), where for the first group a is close to 1 while in the second group a gets closer to 0. In this particular case of constant interests, the solution of Eq. (1) attains a less complex form than in Scenario 1. However, we will use again the finite difference approximation of Eqs. (5), (6), where the definitions of \(M_1(t),~M_2(t)\) are based on constant interests adapted for the two groups correspondingly, to get more intuition about the derived convergence in the number of infected nodes. Specifically, as it will be verified via simulation and numerical results in "Simulation and numerical results without applying control", a higher constant interest by users implies convergence of the number of infected nodes to a higher value.
Scenario 3: increasing vs. decreasing users’ interest
Again, we will use the finite difference approximation of Eqs. (5), (6), where the definitions of \(M_1(t),~M_2(t)\) for each class correspondingly are based on Eq. (7).
Optimal control framework for information diffusion
In this section, we introduce an optimal control framework for the previously presented information diffusion model for a specific class m. The objective in this optimal control problem is to maximize the number of infected (informed) nodes for a topic/class m by applying an exogenous aid/force, i.e., the control, while taking into account associated control costs, e.g., advertising cost. The motivation behind this is twofold. First it might be necessary to apply a control action to boost users’ interest to increase information spreading. Secondly, more resources might be required (by increasing a control signal) when users are more interested in a topic to conserve resources by not wasting them when users are not interested in the propagated information. Thus, this approach will allow affecting the information diffusion over the susceptible (noninformed) users, via properly controlling user interests.
The next proposition allows us to ignore the state constraints expressed in Eq. (11) in the rest of the analysis.
Proposition 1
For any \(u(.)\in \Omega\) , the state function \(I^m(.):[0,T]\rightarrow \mathfrak {R}\) that satisfies \(I^m(0)=I_0^m\) , also satisfies Eq. (11).
Proof

If \(I^m(t_0)=0\) then \(S^m(t_0)=N\) and \(\frac{\mathrm{d} I^m(t)}{\mathrm{d} t}_{t=t_0^{+}}=0\), meaning that \(I^m(t)=0\), for every \(t>t_0\), \(t\le T\).

If \(I^m(t_0)=N\) then \(S^m(t_0)=0\). Thus, \(\frac{\mathrm{d} I^m(t)}{\mathrm{d} t}_{t=t_0^+}=q N \, f_2^{\mathrm{avg}}(t_0)g_2(u(t_0))<0\), since \(f_2^{\mathrm{avg}}(t_0),g_2(u(t_0))>0\), meaning that \(I^m(t_0^+)\le N\). Similarly for all other \(t^{\prime } \in (t_0,T]\) where \(I^m(t^{\prime })=N\).
The following proposition proves that the number of infected nodes for class m, \(I^m(t)\) is strictly positive for every \(t \in [0,T]\).
Proposition 2
We have that \(I^m(t)\ge I_0^m e^{q f_{2_{\max }}^{avg}g_2(u_{\min })t}\ge 0\), \(\forall t\in [0,T]\).
Proof
It holds that \(\frac{\mathrm{d} I^m(t)}{\mathrm{d} t}\ge q I^m(t)f_2^{\mathrm{avg}}(t)g_2(u(t))\), which means that \(\frac{I^m(t)^{\prime }}{I^m(t)}\ge q f_{2{\max }}^{\mathrm{avg}}g_2(u_{\min })\), where \(f_{2\max }^{\mathrm{avg}}=\max _{\forall t} f_2^{\mathrm{avg}}(t)\) and since \(g_2\) is decreasing with u. Thus, \(\ln I^m(t)\ge q f_{2_{\max }}^{\mathrm{avg}} g_2(u_{\min })t+\ln I^m(0)\), yielding \(I^m(t)\ge I_0^m e^{q f_{2_{\max }}^{\mathrm{avg}} g_2(u_{\min }) t}\), for every \(t\in [0,T]\). \(\square\)
Definition 1
The pair \((I^m(.),u(.))\) is an admissible pair if the following hold: (i) \(u(.)\in \Omega\), (ii) the state \((I^m(.))\) constraint of Eq. (10) holds. Then u(.) is called an admissible control [20].
Definition 2
An admissible control u(.) is an optimal control, if \(J(u(.))\ge J(\underline{u}(.))\) for all admissible controls \(\underline{u}(.)\) [20].
Proposition 3
We have that \(\lambda (t)>0\) for \(t\in [0,T)\).
Proof
We follow a similar proof to the one of Lemma 2 in [20]. First we show that \(\lambda (t)\) is strictly positive over an interval of nonzero length towards the end of [0, T). It holds that \(\lambda (T) = k_I \ge 0\). If \(k_I > 0\), this statement holds due to continuity. If \(k_I = 0\), then from (13) and for \(t=T\) we have: \(\frac{d\lambda (t)}{dt}_{t=T} =  k_1 <0\), i.e., descending from positive values before reaching the value \(k_I=0\), and this statement also holds.
As \(t^{\prime }< T\), consider the latest time in [0, T) that \(\lambda (t^{\prime })=0\), i.e., \(\lambda (t)>0\) for \(t^{\prime }<t<T\). Then, \(\frac{d\lambda (t)}{dt}_{t=t^{{\prime }+}} = k_1 < 0\) which is impossible since for \(t>t^{\prime }\), \(\lambda\) is positive and thus it should increase from the zero value. The latter statement concludes the proof of Proposition 3. \(\square\)
 1.\(g_1\) convex and \(g_2\) concave. Then \(\phi\) is convex with respect to u. Due to convex maximization, the optimal control will be necessarily at the extrema of the range of the control, determined by comparison as:$$\begin{aligned} u^* = \left\{ \begin{array}{rl} u_{\min } &{} \text{ if } \phi (u_{\min })>\phi (u_{\max }), \\ u_{\max } &{} \text{ if } \phi (u_{\min })<\phi (u_{\max }). \end{array} \right. \end{aligned}$$(17)
 2.\(g_1\) concave and \(g_2\) convex. Then \(\phi\) is concave with respect to u. In this case, a concave maximization takes place, where the maxima of \(\phi (u)\) occur at the points where the partial derivative with respect to u is zero, or at the extrema of the range of the control, determined by comparison. The equation \(\frac{\partial \phi }{\partial u}=0\) becomes:where \(\lambda >0\) from Proposition 3. If \(u^{\prime }\) is the solution of the above, then the optimal control becomes:$$\begin{aligned} \frac{k_2}{\lambda } &= \frac{\partial g_1(u)}{\partial u} \left[ \frac{N_s^{\mathrm{avg}}p_1}{N} f_1^{\mathrm{avg}}f_3^{\mathrm{avg}} (NI^m) I^m \right. \nonumber \\ & \left. + \,\, p_2 \frac{\pi R^2}{L^2} f_{1}^{\mathrm{avg}} I^m (NI^m)\right] \nonumber \\ & \frac{\partial g_2(u)}{\partial u} q I^m f_{2}^{\mathrm{avg}}, \end{aligned}$$(18)In this case based on the explicit forms of functions \(g_1,~g_2\), we can study possible relations/properties of the optimal control with respect to users’ interests, as it will be performed in the following sections.$$\begin{aligned} u^*=\max \left\{ u_{\min },\min \{u^{\prime },u_{\max }\}\right\} . \end{aligned}$$(19)
 3.\(g_1\) concave and \(g_2\) concave. We havewhere \(A_1 = \frac{N_s^{\mathrm{avg}}p_1}{N} f_1^{\mathrm{avg}}f_3^{\mathrm{avg}} (NI^m) I^m \frac{\partial ^2 g_1(u)}{\partial u^2}+p_2 \frac{\pi R^2}{L^2} f_{1}^{\mathrm{avg}}I^m (NI^m) \frac{\partial ^2 g_1(u)}{\partial u^2}\le 0\) and \(B_1 = q I^m f_{2}^{\mathrm{avg}}\frac{\partial ^2 g_2(u)}{\partial u^2}\le 0\). Thus, \(\frac{\partial ^2 \phi (u)}{\partial u^2}\le 0\) if \(A_1\ge B_1\), that leads to a concave maximization as in case 2 above, or \(\frac{\partial ^2 \phi (u)}{\partial u^2}\ge 0\) if \(A_1\le B_1\), that leads to a convex maximization as in case 1 above.$$\begin{aligned} \frac{\partial ^2 \phi (u)}{\partial u^2} &= \lambda \left[ \frac{N_s^{\mathrm{avg}}p_1}{N} f_1^{\mathrm{avg}}f_3^{\mathrm{avg}} (NI^m) I^m \frac{\partial ^2 g_1(u)}{\partial u^2}\right. \nonumber \\ &\left. +p_2 \frac{\pi R^2}{L^2} f_{1}^{\mathrm{avg}}I^m (NI^m) \frac{\partial ^2 g_1(u)}{\partial u^2}\right. \nonumber \\ &\left. \vphantom {\frac{\partial ^2 g_1(u)}{\partial u^2}}  q I^m f_{2}^{\mathrm{avg}}\frac{\partial ^2 g_2(u)}{\partial u^2}\right] , \end{aligned}$$(20)
 4.
\(g_1\) convex and \(g_2\) convex. Then \(\frac{\partial ^2 g_1(u)}{\partial u^2}\ge 0\), \(\frac{\partial ^2 g_2(u)}{\partial u^2}\ge 0\). Thus, \(\frac{\partial ^2 \phi (u)}{\partial u^2}\ge 0\) if \(A_1\ge B_1\), that leads to a convex maximization as in case 1 above, or \(\frac{\partial ^2 \phi (u)}{\partial u^2}\le 0\) if \(A_1\le B_1\), that leads to a concave maximization as in case 2 above.
We should note that the controller applies one kind of control with aim to increase \(I^m(t)\) tradingoff cost, but this impacts in a different way each part of the information propagation equation (i.e., Eq. 10) via the functions \(g_1(u),g_2(u)\).
Obviously, \(\Gamma\) is a concave function of \(I^m\), attaining its maximum at \(I^m_{\max }=\frac{F R_{\mathrm{avg}}^m+ G (\frac{1}{R_{\mathrm{avg}}^m}1)+K}{D R_{\mathrm{avg}}^m+E}\), where \(F=\frac{N_S^{\mathrm{avg}} p_1}{2M}\), \(E=\frac{p_2 \pi R^2}{L^2 M}\), \(D=\frac{N_S^{\mathrm{avg}} p_1}{MN}\), \(G=\frac{q}{2M}\), \(K=\frac{p_2 \pi R^2N}{L^2 2M}\). Note that \(I^m_{\max }\) is decreasing with interest when \((FE +GDDK)(R_{\mathrm{avg}}^{m})^22GD R_{\mathrm{avg}}^mGE <0\), where D, E, F, G, K, are determined by the parameters of the system. In this case, for higher values of interest it is intuitively expected that the optimal control (Eq. 22) will achieve its maximum on a lower value of \(I^m\), if ignoring any dependence of \(\lambda\) on the examined parameters, i.e., \(I^m, R_{\mathrm{avg}}^m\).
Computing the optimal control value
Although Eqs. (17, 19) provide the form of the optimal control, computing the optimal control value is more complex, demanding the knowledge of the value of the adjoint variable, \(\lambda\), for each t. In this section, we construct the Hamilton–Jacobi–Bellman (HJB) equation [27–29] and solve it via a numerical approach to obtain optimal control values within the control time interval ([0, T]).
Definition 3
Simulation and numerical results without applying control
In this section, we present simulation and numerical results for each users’ interest scenario of "Information diffusion modeling and analysis without control". Specifically, the simulation results refer to the realization of the diffusion model described in "Information diffusion modeling and analysis without control" in MATLAB, while numerical results refer to the approximate solution (via finite difference scheme) of the ODEs in each scenario with the same parameters as in the corresponding simulation.
The simulation setting is as follows. We consider a generalized network, the wireless substrate of which consists of a wireless multihop network with \(N=500\) nodes deployed over a square region with side \(L=350m\) and with homogeneous transmission radius among nodes equal to \(R=25m\). All simulation results are obtained as averages over several wireless topologies (\(\#2\)) and multiple repetitions (\(\# 3\)) for the diffusion at each topology. Furthermore, we examine two overlaying social network topologies over the same set of nodes as the wireless substrate, namely one scalefree and one smallworld [32]. For the scalefree network topology, the social degree for each node is drawn from the powerlaw distribution with exponent 3, as observed for many social networks [33] (specifically the probability density function is \(f(x)=\left( \frac{2}{x}\right) ^{3},~x\ge 2\)), and the corresponding social layer’s neighbors of each node are chosen randomly. The smallworld topology is constructed following the Watts & Strogatz paradigm [34]. For both topologies, \(N_S^{\mathrm{avg}}\cong 4\). However, the degree distribution in the smallworld topology is much more homogeneous than the corresponding one of the scalefree social topology. Also, the scalefree topology presents low average path length, which is a smallworld feature [32]. Note that the value \(\Delta t\) should be appropriately small, so that the solution of the ODE derived via the finite difference scheme (Eqs. 5, 6), approximates closely the precise solution of the ODE. We chose \(\Delta t=0.4\). Finally, 10 nodes out of 500 are initially infected (e.g., via MMS) for each class in all simulation and numerical results that follow.
Scenario 1: information diffusion dynamics in the case of periodic users’ interests
In this scenario, we consider that the users’ interests vary uniformly and randomly with mean value \(1\frac{1}{4}\sin ^2(\frac{\pi }{180}(t+100))\frac{1}{7}\) for the first class and \(\frac{1}{4}\sin ^2(\frac{\pi }{180}(t+100))+\frac{1}{7}\) for the second class with the constraint that the interests of one node for the two classes are complementary (i.e., in the corresponding scenario of "Information diffusion modeling and analysis without control", \(A=\frac{1}{4},~B=\frac{1}{7},~a=\frac{\pi }{180},~b=100\)). Therefore, the period of the interests’ functions is one year, a fact that can be reflected in realistic situations as the ones explained in "Background". The values of \(p_1,~p_2,~q\) will be specified in each simulation case.
Fig. 2a compares the dynamics of the information diffusion for class 1 as derived by numerically solving Eq. (3) with the results obtained via simulations according to the proposed diffusion model in "Information diffusion modeling and analysis without control". The same is illustrated in Fig. 2b for class 2 (where the numerical results are obtained via numerically solving Eq. 4). The involved parameters take the values \(p_1=p_2=0.5\), i.e., P and MMS types of transfer take place with the same probability in case of diffusion and \(q=0.2\). The results are the same independently of the social topology, i.e., smallworld or scalefree.
To conclude for this scenario, the theoretical model overestimates the volume of the information spreading, especially for class 2, which is characterized by lower values of interest. Also, the behavior in both cases of social topologies, i.e., smallworld and scalefree, does not differentiate significantly.
Scenario 2: information diffusion dynamics in the presence of groups with different characteristics
Scenario 3: information diffusion dynamics in the case of increasing vs. decreasing users’ interests
Simulation results, numerical results and discussion in controlled information diffusion
In this section, we further study and evaluate the introduction of control—as described in "Optimal control framework for information diffusion"—in the three scenarios of "Simulation and numerical results without applying control". The values of the parameters that are used in "Simulation and numerical results without applying control" remain the same, except otherwise mentioned. Additionally, we consider \(u_{\min }=0,~u_{\max }=30\), \(\Delta I^m=1,~\forall ~m\), \(\Delta t=10^{4}\), \(k_1=1,~k_2=3,k_I=1\), \(T=2\), and finally, the topology on the social layer is considered as scale free. Note that \(\Delta t<< \Delta I^m\) so that the HJB solution converges [31]. The control is applied to only one class (or equivalently one group for constant interests) and specifically, we chose the class \(m=2\) (or equivalently Group 2 for constant interests), to evaluate how information diffusion behaves under low values of interest when introducing control, and compare this behavior with the case when no control is applied (similarly to "Simulation and numerical results without applying control"). In the following subsections, in each scenario, we compare the numerical (derived via Eq. 10 using Eq. 21) and simulation results with and without control regarding the number of infected nodes for the second class/group, while we also study several properties and the behavior of the optimal control itself. We adapt the diffusion model of "Information diffusion modeling and analysis without control" to introduce control by replacing the probabilities \(f_1^{\mathrm{avg}}(t),~ f_2^{\mathrm{avg}}(t)\) with \(f_1^{\mathrm{avg}}(t) g_1(u(t)),~f_2^{\mathrm{avg}}(t) g_2(u(t))\), as implied by comparing Eq. (10) with Eq. (1). Every simulation runs for 200 time steps, i.e., the control time \(T=2\) is divided into smaller time intervals each having a duration of 0.01.
Scenario 1: controlled information diffusion dynamics in the case of periodic users’ interests
Similar to "Scenario 1: information diffusion dynamics in the case of periodic users’ interests", we consider that the users’ interests vary uniformly and randomly with mean value \(1\frac{1}{4}\sin ^2(\frac{\pi }{0.5}(t+0.2))\frac{1}{7}\) for the first class and \(\frac{1}{4}\sin ^2(\frac{\pi }{0.5}(t+0.2))+\frac{1}{7}\) for the second class with the constraint that the interests of one node for the two classes are complementary.
Scenario 2: controlled information diffusion dynamics in the presence of groups with different characteristics
In this section, we consider constant users’ interests adopting the same parameter values as in "Scenario 2: information diffusion dynamics in the presence of groups with different characteristics". Fig. 10 presents the dynamics of the information diffusion for Group 2 and indicates several properties of the optimal control. Specifically, Fig. 10a depicts the dynamics of information diffusion for Group 2 derived from simulations and computed numerically both cases with and without control (Eqs. 1, 10). As in the case of periodic users’ interests, the introduction of control increases the number of infected nodes. Note that the numerical results of Eq. (10) refer to a time interval of length 2 after convergence (see also the discussion in "Scenario 1: information diffusion dynamics in the case of periodic users’ interests"). As it is shown in Fig. 10a, introducing control leads to a tighter approximation of the simulation results from the numerical ones, compared to the absence of control.
Scenario 3: controlled information diffusion dynamics in the case of decreasing with time users’ interests
In this section, we consider users’ interests that decrease with time using the same parameters/interest functions with "Scenario 3: information diffusion dynamics in the case of increasing vs. decreasing users’ interest". Figure 11 presents the dynamics of the information diffusion for class 2 and indicates several properties of the optimal control. Specifically, Fig. 11a depicts the dynamics of information diffusion for the second class (\(m=2\)) derived from simulations and computed numerically both cases with and without control (Eqs. 1, 10). We observe that the decrease of users’ interests is reflected to the dynamics of information diffusion, although the later exhibits a much smaller decreasing rate. As in the cases of periodic and constant users’ interests, the introduction of control increases the number of infected nodes. Note that the numerical results of Eq. (10) refer to a time interval of length 2 after convergence (see also the discussion in "Scenario 1: controlled information diffusion dynamics in the case of periodic users’ interests"). As it is shown in Fig. 11a, introducing control leads to a tighter approximation of the simulation results from the numerical ones compared to the absence of control.
Conclusions
In this paper, we introduced a novel framework for modeling and controlling useful information diffusion in generalized networks that takes into account user interests and their features, i.e., interest periodicity or interest dependence on the topic of the propagated information. The epidemic equations were numerically solved and compared with simulation results for three indicative operational scenarios, yielding significant results on the impact of each associated factor (e.g., topology layer, interest values and time variedness) on diffusion dynamics. Furthermore, optimal controls were obtained and studied over each information class, while simulation and numerical results are provided for the cases where user interest is low and diffusion needs boosting to improve the efficiency of useful information spreading. Interesting behavioral properties of the optimal controls with respect to their dependence on the evolving users’ interests and the number of infected nodes are shown via analysis (on an intuitive basis) and numerical evaluations. Our future work will focus on studying the cases where interest classes may have correlations, and the impact that these correlations may have on the corresponding controls.
Declarations
Authors’ contributions
ES contributed to the development and analysis of the proposed framework for modeling and studying information diffusion. She also developed the evaluation part and participated in the manuscript writing. VK contributed in the development and analysis of the framework, participated in the design of the study, in the sequence alignment and contributed in the manuscript writing. SP participated in the design of the study and its coordination. All authors read and approved the final manuscript.
Acknowledgements
This research is cofinanced by the European Union (European Social Fund) and Hellenic national funds through the Operational Program ’Education and Lifelong Learning’ (NSRF 20072013).
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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