CN108667833A - Communication system Malware based on coupling propagates modeling and method for optimally controlling - Google Patents
Communication system Malware based on coupling propagates modeling and method for optimally controlling Download PDFInfo
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Abstract
The invention discloses the Malwares based on coupling to propagate modeling and method for optimally controlling, includes at least：S1 considers the unidirectional couplings between two the Malware A and B propagated simultaneously in computer network, and the respective Dynamical model of Malware A and B is established respectively using SIS models；S2 builds cost functional, and using the respective Dynamical model of Malware A and B as constraint function using artificial removal rate as control variable；S3 combining targets functional and constraint function solve optimal control variable in given domination set.Through simulating, verifying, the method for the present invention can also make control cost maintain reduced levels while inhibiting Malware to propagate.
Description
Technical field
The present invention relates to Communication and Information Systems and network control system to model field, more particularly to logical based on coupling
Letter system Malware propagates modeling and method for optimally controlling.
Background technology
Malware (Malware) spreads through the internet in communication system, and huge loss is brought to the mankind.It is more preferable
Understanding these Malwares the mechanism of transmission, find suitable method and inhibit its propagation, Recent study person are for difference
The Malware of Spread type establishes the propagation model of some Malwares with different visual angles, and based on this deeply
Ground analyzes Spread type and propagation characteristic.
In research, a kind of most representational method is to be disliked by epidemic transmission compartment model to study computer
The propagation of meaning software.Epidemic disease compartment model analyzes the propagation with transmissible disease originating from people.Common warehouse mould
Type has SI (Susceptible-Infected), SIS (Susceptible-Infected-Susceptible) and SIR
(Susceptible-Infected-Remove).If an individual is in S warehouses, then it represents that it is in health status.And I
Indicate Infection Status and removal state respectively with R.Compartment model and its analysis method are completely suitable on research computer network
The Communication Research of Malware.Current numerous studies both domestic and external are opened on the basis of above three classics compartment model
Exhibition is established rational mathematical model and is analyzed by the different factors for considering to influence to propagate.
Past research, the propagation mainly for some Malware are analyzed.Actually a variety of Malwares are same
When propagate and be widely present on the internet, such as macrovirus is while a computer causes damage, and trojan horse is also
It hides and carries out certain illegal operations in this computer.It will be apparent that when having multiple Malwares while propagating, it is relatively simple
Mode be to separate the propagation of two kinds of Malwares, it is believed that their communication process is independent from each other.This hypothesis letter
Analysis is changed, but its result tends not to the actual conditions for reflecting multiple Malwares while propagating.
In recent years, begin with some for it is a variety of coupling infectious diseases epidemic transmission problems, primarily directed to two or
The propagation of the multiple biological virus of person conducts a research.The result of these researchs obviously can be generalized to computer malware propagation
In analysis, however current research is all that research is unfolded just for a certain coupled relation, this makes the universal of model
It is poor.In general, the coupled relation between a variety of Malwares can be mutually promoted, can also be resource of vying each other causes to propagate
On mutual inhibition, it is also possible to be by the initial inhibition promoted finally.
In conjunction with this concept, present invention proposition is described using a nonlinear function in two kinds of Malware communication processes
All couplings give a unified research frame for the communication process modeling and analysis of two kinds and a variety of Malwares
Frame.
Invention content
The object of the present invention is to provide the communication system Malwares based on coupling to propagate modeling and method for optimally controlling.
Malware provided by the invention based on coupling propagates modeling and method for optimally controlling, includes at least：
S1 considers the unidirectional couplings between two the Malware A and B propagated simultaneously in computer network, utilizes SIS models point
The respective Dynamical model of Malware A and B is not established；According to the node individual amount for being uninfected by a Malware and it is somebody's turn to do
Density of infection caused by one Malware meets normalizing condition, determines the feasible zone of Malware A and B；
S2 is with artificial removal rate δ1(t) and δ2(t) as control variable, with SA(t)、IA(t)、SB(t)、IB(t) it is state
Variable builds cost functionalAnd with Malware A and
The respective Dynamical models of B are constraint function；
Wherein, t indicates that moment, t ∈ [0, T], [0, T] are given time range；SA(t) and SB(t) moment is indicated respectively
The node individual amount of Malware A and B are uninfected by when t；IA(t) and IB(t) indicate that Malware A and B make when moment t respectively
At density of infection；δ1(t) and δ2(t) the artificial removal rate of Malware A and B when moment t are indicated respectively；c1And c2Table respectively
Show income and the weight of consumption；
S3 combining targets functional and constraint function solve optimal control variable in given domination set.
Further, in step S1, the Dynamical model of the Malware A established is：
The Dynamical model of the Malware B established is：
Wherein：T indicates the moment；SA(t) and SB(t) the node individual that Malware A and B are uninfected by when moment t is indicated respectively
Quantity；IA(t) and IB(t) density of infection caused by Malware A and B when moment t is indicated respectively；<k>Indicate computer network
Average degree；γ1And γ2Indicate node clear-cutting forestland rate；δ1(t) and δ2(t) people of Malware A and B when moment t are indicated respectively
Work removal rate；β1(t) and β2(t) the time-varying infection rate of Malware A and B, β are indicated respectively1(t) ∈ (0,1], β2(t)∈(0,
1]；
The definition of the time-varying infection rate is：Wherein：
WithIndicate not considering that Malware A and B is respectively being propagated through when Malware A and B influence each other respectively
Infection rate in journey,WithFor empirical value；
α1(t) and α2(t) coupling terms are indicated,
α2(t)=1； It is the critical value for describing coupling between Malware B and A, is empirical value, by repeatedly trying
Test determination.
Further, in step S1, the feasible zone Ω of Malware A and B are：
Wherein, SA(t) and SB(t) the node individual amount that Malware A and B are uninfected by when moment t is indicated respectively；IA(t)
And IB(t) density of infection caused by Malware A and B when moment t is indicated respectively；Indicate the positive real number domain of 2 dimensions.
Further, in step S2, the constraint function is as follows：
Wherein：T indicates the moment；SA(t) and SB(t) the node individual that Malware A and B are uninfected by when moment t is indicated respectively
Quantity；IA(t) and IB(t) density of infection caused by Malware A and B when moment t is indicated respectively；<k>Indicate computer network
Average degree；γ1And γ2The Dynamical model interior joint clear-cutting forestland rate of Malware A and B are indicated respectively；δ1(t) and δ2
(t) the artificial removal rate of Malware A and B when moment t are indicated respectively；WithIt indicates not considering Malware A and B respectively
Infection rates of the Malware A and B in respective communication process when influencing each other,WithFor empirical value；
It is the critical value for describing coupling between Malware B and A, is empirical value, is determined by test of many times.
Further, step S3 further comprises：
The Lagrangian of 310 structure optimal control problems
320 according to LagrangianL constructor H：
330 analyze optimal control problem using Pontryagin maximal principles, obtain adjoint variable λ1(t)、λ2(t)、λ3
(t)、λ4(t) should meet：
340 combine transversality condition λ1(T)=λ2(T)=λ3(T)=λ4(T)=0 optimal control variable, is calculated, it is as follows：
Wherein：
IA(t) and IB(t) density of infection caused by Malware A and B when moment t is indicated respectively；δ1(t) and δ2(t) respectively
Indicate the artificial removal rate of Malware A and B when moment t；c1And c2The weight of income and consumption is indicated respectively；<K indicates to calculate
The average degree of machine network；γ1And γ2Indicate node clear-cutting forestland rate；SA(t) and SB(t) it indicates to be uninfected by evil when moment t respectively
The node individual amount of meaning software A and B；λ1(t)、λ2(t)、λ3(t)、λ4(t) adjoint variable when moment t is indicated；With
Indicate not considering infection rates of the Malware A and B in respective communication process when Malware A and B influence each other respectively,WithFor empirical value； It is the critical value for describing coupling between Malware B and A, is empirical value,
It is determined by test of many times；T indicates particular moment constant when Lagrange multiplier is 0；Indicate optimal state variable SA(t)、IA(t)、SB(t)、IB(t)；WithFor [0,
1] arbitrary constant in indicates the upper bound of controlled variable.
Compared to the prior art, the invention has the advantages that and advantageous effect：
(1) the unidirectional couplings effect expansion research between two kinds of Malwares propagated simultaneously, is two kinds of Malwares
Communication process provides a Unified frame simultaneously.
(2) consider the unidirectional couplings between two the Malware A and B propagated simultaneously in computer network, it is dynamic to construct propagation
Mechanical model, and propose the optimal control problem based on Dynamical model；Through simulating, verifying, optimum control side of the present invention
Method can also significantly reduce control cost under the premise of ensuring that infected number of nodes is as few as possible.
(3) it is suitable for the propagation of rumour, the propagation of biological virus and the fault propagation in electric system, it is versatile.
Description of the drawings
Fig. 1 is the density of infection tendency chart of infected with malware A under different control strategies；
Fig. 2 is the density of infection tendency chart of infected with malware B under different control strategies；
Fig. 3 is optimum control variableWithCorresponding infection node ratio trend chart.
Specific implementation mode
In order to illustrate more clearly of the present invention and/or technical solution in the prior art, below originally by control description of the drawings
The specific implementation mode of invention.It should be evident that drawings in the following description are only some embodiments of the invention, for this
For the those of ordinary skill of field, without creative efforts, others are can also be obtained according to these attached drawings
Attached drawing, and obtain other embodiments.
It should be appreciated that the specific embodiments described herein are merely illustrative of the present invention, it is not intended to limit the present invention.
In addition, technical characteristic involved in the various embodiments of the present invention described below is as long as they do not conflict with each other
It can be combined with each other.
The specific implementation mode of the present invention is illustrated below in conjunction with attached drawing.
The present invention is based on the communication system Malwares of coupling to propagate modeling and method for optimally controlling, is as follows：
S1 considers the unidirectional couplings between two the Malware A and B propagated simultaneously in computer network, utilizes SIS models point
The respective Dynamical model of Malware A and B is not established；According to the node individual amount for being uninfected by a Malware and it is somebody's turn to do
Density of infection caused by one Malware meets normalizing condition, determines the feasible zone of Malware A and B.
The specific implementation process of this step is provided below.
Consider two Malware A and B in some computer network while propagating, it is assumed that the average degree of network is<k>,
Under the frame of mean field, the Dynamical model of Malware A and B is established, it is as follows：
Formula (1) is the Dynamical model of Malware A, and formula (2) is the Dynamical model of Malware B.Formula
(1) in~(2)：
T indicates the moment；
SA(t) the node individual amount that Malware A is uninfected by when moment t is indicated；
SB(t) the node individual amount that Malware B is uninfected by when moment t is indicated；
IA(t) and IB(t) density of infection caused by Malware A and B when moment t is indicated respectively；The density of infection refers to meter
The number of nodes of infected with malware accounts for the ratio of all number of nodes in calculation machine network；
γ1And γ2The clear-cutting forestland rate of the Dynamical model interior joint of Malware A and B is indicated respectively；
δ1(t) and δ2(t) the artificial removal rate of Malware A and B when moment t are indicated respectively；
β1(t) and β2(t) the time-varying infection rate of Malware A and B are indicated respectively, wherein β1(t) ∈ (0,1], β2(t)∈
(0,1]。
The calculating of time-varying infection rate is as follows：
Wherein,WithIt indicates not considering Malware A and B phase respectively
Infection rates of the Malware A and B in respective communication process when being influenced between mutually,WithIt is true by test of many times for empirical value
It is fixed；α1(t) and α2(t) it is coupling terms, is respectively used between description Malware A and B and the coupling between B and A.
For ease of analysis, the present invention considers the unidirectional couplings between Malware A and B, even α2(t)=1.
Enable α1(t) it is defined as follows：
In formula (3), It is the density of infection critical value for describing coupling between Malware B and A, is
Empirical value is determined by test of many times.
Above-mentioned α1(t) definition is to work as I based on considered belowB(t)=0 when, without coupling between Malware B and A,
Therefore α1(t)=1.
On the other hand, it is contemplated that β1(t) ∈ (0,1], therefore α1(t) there is the upper boundWhenWhen, α1(t) will
It is not up to this upper bound, at this time if IB(t) ≠ 0, so that it may be write as coupling termsIt will be apparent that coupling
The definition for closing item includes all unidirectional couplings effects between B and A.
Assuming that SA(t)、IA(t)、SB(t)、IB(t) meet normalizing condition, i.e. SA(t)+IA(t)=SB(t)+IB(t)=1,
Then the feasible zone Ω of Malware A and B are in computer network：
In formula (4),Indicate the positive real number domain of 2 dimensions.
Getting off will be directed toThe case where the specific implementation process of step S2~S4 is provided,
In, indicate do not include.
S2 is with the artificial removal rate δ of Malware A and B1(t) and δ2(t) it is used as and controls variable, structure cost functional, and with
The respective Dynamical models of Malware A and B are constraint function.
In the Dynamical model of Malware, artificial removal rate δ1(t) and δ2(t) it as unique control variable, gives
The set of fixed following artificial removal rate is as domination set：
In formula (5), t indicates the moment；T ＞ 0 are a given time constant, L2(0, T) two-dimensional integral is indicated；With
To belong to the arbitrary constant of [0,1], the upper bound of control variable is indicated.
In order to keep the number of infected node minimum by control, and the consumption of communication system is minimum, considers such as
Lower cost functional J, constraint function and primary condition, cost functional are shown in that formula (6), constraint function are shown in formula (7), and primary condition is shown in formula
(8)：
In formula (6)~(8)：
IA(t) and IB(t) it indicates to infect caused by Malware A and B respectively in computer network when moment t respectively close
Degree；
δ1(t) and δ2(t) it is expressed as reducing IA(t) and IB(t) cost paid, the i.e. people of Malware A and B
Work removal rate such as removes the cost of Malware A and B；
c1And c2The weight for indicating income and consumption respectively is system given value；
S (0) and I (0) indicates original state；
S0Indicate that 0 moment was uninfected by the node individual amount of Malware；
I0Density of infection amount caused by indicating 0 moment Malware.
Above-mentioned constraint function (7) it is rewritable at：
In formula (9)：
φ is indicated by SA(t)、IA(t)、SB(t)、IB(t) vector constituted,
B is coefficient matrix,
Therefore have：
Wherein, φ1And φ2Indicate two groups of different state vectors, " ' " on target indicate φ1Corresponding parameter, band " ' " on
Target indicates φ2Corresponding parameter.
Then：
Wherein
Therefore it obtains：
Therefore constant V=max M, | | B | | ＜ ∞, | | | | representing matrix norm.
Therefore function D (φ) meets the Li Puxizi conditions of continuity, from the definition of control variable and to state variable SA(t)、IA
(t)、SB(t)、IB(t) limitation, it can be deduced that the solution of constraint function exists.
The final purpose of cost functional is to obtain optimal control variableMake its satisfaction
S3 is according to there are optimal control variablesCost functional is set to set up, it is optimal according to providing
Optimum state solution under control system and corresponding constraint function, primary condition, then there are adjoint variables to meet condition, in addition
Optimum control condition is obtained in conjunction with given domination set.
Provide Lagrangian (Lagrange) the functions L of optimal control problem：
In formula (14), IA(t) and IB(t) computer network density of infection caused by Malware A and B is indicated respectively；δ1
(t) and δ2(t) it represents and reduces IA(t) and IB(t) cost paid；c1And c2The weight of income and consumption is indicated respectively,
For system given value.
Define Hamiltonian (Hamilton) function H：
In formula (15), λ1(t)、λ2(t)、λ3(t)、λ4(t) adjoint variable of moment t is indicated.
There are optimal control variablesFormula (13) is set up, and meets constraint function (see formula
And primary condition (7)) (see formula (8)).
Pontryagin (Pang Te lia kings) maximal principle is used to analyze optimal control problem below, by Pontryagin
Maximal principle provides optimal control variable systemWith constraint function, the optimum state variable of primary conditionSo have, adjoint variable λ1(t)、λ2(t)、λ3(t)、λ4(t) should meet：
Transversality condition：
In formula (17), T indicates particular moment constant when Lagrange multiplier is 0.
In addition, having：
In formula (18),WithFor arbitrary constant in [0,1], the upper bound of controlled variable is indicated.
S4 carries out numerical simulation verification using MATLAB.
Using MATLAB platforms, chooses suitable parameter and establish the Malware propagation model based on coupling, pass through ratio
Compared with the variation tendency for infecting node under different situations, to verify the superiority of method for optimally controlling of the present invention.
In numerical simulation, optimum control variable system can carry out numerical solution with Euler method.Consider a stochastic computer net
Network, number N=1000 of node, the average degree of network<k>=6, given primary condition is IA(0)=0.05, IB(0)=
0.05；Through test of many times, other parameters are chosen：γ1=0.01, γ2=0.02, c1=2, c2=
1, Choose the time T=300 of optimum control.
Fig. 1 and Fig. 2 gives Malware A and B respectively in no control, constant value control, feedback control and optimal control
Make the variation tendency that node is infected under (i.e. the method for the present invention).It, must under the parameter of above-mentioned setting for not having controlled situation
The outburst of endemic disease is so had, i.e. Malware can be spread in the entire network.And other three kinds of control strategies can be controlled effectively
The propagation of related object processed, and constant value control and optimum control can make the number of nodes of infected with malware be reduced to 0.
In order to better illustrate the superiority of the method for the present invention, above-mentioned four kinds of control strategies are calculated separately in different terminals
The totle drilling cost when moment, related data are shown in Table 1.The lowest cost of optimal control policy is used as can be seen from the table.It is setting
Under fixed parameter, optimum control and constant value control can make the number of infection node be reduced to 0, it is obvious that totle drilling cost
Totle drilling cost table of the 1 four kinds of control strategies of table in different terminal junctures
Optimal control policy achievees the purpose that controlling Malware propagates by the number of infection control node, when its propagation
After centainly being controlled, need to take the number of nodes of control to reduce gradually, optimum control variableWithCurve see Fig. 3
It is shown.
Present embodiment is demonstrated mainly for the unidirectional couplings effect between two kinds of Malwares, is proposed
For two object communication processes propagation model to the failure in the propagation of rumour, the propagation of biological virus and electric system
It propagates equally applicable.
Specific embodiment described herein is only to be given an example to patent spirit of the present invention.Patent institute of the present invention
Belonging to those skilled in the art can make various modifications or additions to the described embodiments or using similar
Mode substitute, but without departing from the spirit or beyond the scope defined by the appended claims of patent of the present invention.
Claims (5)
1. the Malware based on coupling propagates modeling and method for optimally controlling, characterized in that include at least：
S1 considers the unidirectional couplings between two the Malware A and B propagated simultaneously in computer network, is built respectively using SIS models
The respective Dynamical models of vertical A and B；It is made according to the node individual amount and a Malware that are uninfected by a Malware
At density of infection meet normalizing condition, determine the feasible zone of A and B；
S2 is with artificial removal rate δ1(t) and δ2(t) as control variable, with SA(t)、IA(t)、SB(t)、IB(t) it is state variable,
Build cost functionalAnd respectively with Malware A and B
Dynamical model be constraint function；
Wherein, t indicates that moment, t ∈ [0, T], [0, T] are given time range；SA(t) and SB(t) when indicating moment t respectively
It is uninfected by the node individual amount of Malware A and B；IA(t) and IB(t) it indicates respectively when moment t caused by Malware A and B
Density of infection；δ1(t) and δ2(t) the artificial removal rate of Malware A and B when moment t are indicated respectively；c1And c2It indicates to receive respectively
The weight of benefit and consumption；
S3 combining targets functional and constraint function solve optimal control variable in given domination set.
2. the Malware based on coupling propagates modeling and method for optimally controlling as described in claim 1, it is characterized in that：
In step S1, the Dynamical model of the Malware A established is：
The Dynamical model of the Malware B established is：
Wherein：T indicates the moment；SA(t) and SB(t) the node number of individuals that Malware A and B are uninfected by when moment t is indicated respectively
Amount；IA(t) and IB(t) density of infection caused by Malware A and B when moment t is indicated respectively；<k>Indicate the flat of computer network
Spend；γ1And γ2Indicate node clear-cutting forestland rate；δ1(t) and δ2(t) indicate that Malware A's and B when moment t is artificial respectively
Removal rate；β1(t) and β2(t) the time-varying infection rate of Malware A and B, β are indicated respectively1(t) ∈ (0,1], β2(t)∈(0,1]；
The definition of the time-varying infection rate is：β1(t)=α1(t)β1 0,Wherein：
β1 0WithIndicate not considering that Malware A and B is in respective communication process when Malware A and B influence each other respectively
Infection rate, β1 0WithFor empirical value；
α1(t) and α2(t) coupling terms are indicated,α2(t)
=1； It is the critical value for describing coupling between Malware B and A, is empirical value, it is true by test of many times
It is fixed.
3. the Malware based on coupling propagates modeling and method for optimally controlling as described in claim 1, it is characterized in that：
In step S1, the feasible zone Ω of Malware A and B are：
Wherein, SA(t) and SB(t) the node individual amount that Malware A and B are uninfected by when moment t is indicated respectively；IA(t) and IB
(t) density of infection caused by Malware A and B when moment t is indicated respectively；Indicate the positive real number domain of 2 dimensions.
4. the Malware based on coupling propagates modeling and method for optimally controlling as described in claim 1, it is characterized in that：
In step S2, the constraint function is as follows：
Wherein：T indicates the moment；SA(t) and SB(t) the node number of individuals that Malware A and B are uninfected by when moment t is indicated respectively
Amount；IA(t) and IB(t) density of infection caused by Malware A and B when moment t is indicated respectively；<k>Indicate the flat of computer network
Spend；γ1And γ2The Dynamical model interior joint clear-cutting forestland rate of Malware A and B are indicated respectively；δ1(t) and δ2(t)
The artificial removal rate of Malware A and B when moment t are indicated respectively；β1 0WithIt indicates not considering that Malware A and B are mutual respectively
Between infection rates of the Malware A and B in respective communication process when influencing, β1 0WithFor empirical value； It is to retouch
The critical value of coupling between Malware B and A is stated, is empirical value, is determined by test of many times.
5. the Malware based on coupling propagates modeling and method for optimally controlling as described in claim 1, it is characterized in that：
Step S3 further comprises：
The Lagrangian of 310 structure optimal control problems
320 according to LagrangianL constructor H：
330 analyze optimal control problem using Pontryagin maximal principles, obtain adjoint variable λ1(t)、λ2(t)、λ3(t)、
λ4(t) should meet：
340 combine transversality condition λ1(T)=λ2(T)=λ3(T)=λ4(T)=0 optimal control variable, is calculated, it is as follows：
Wherein：
IA(t) and IB(t) density of infection caused by Malware A and B when moment t is indicated respectively；δ1(t) and δ2(t) it indicates respectively
The artificial removal rate of Malware A and B when moment t；c1And c2The weight of income and consumption is indicated respectively；<k>Indicate computer network
The average degree of network；γ1And γ2Indicate node clear-cutting forestland rate；SA(t) and SB(t) it indicates to be uninfected by when moment t respectively maliciously soft
The node individual amount of part A and B；λ1(t)、λ2(t)、λ3(t)、λ4(t) adjoint variable when moment t is indicated；β1 0WithRespectively
Expression does not consider infection rates of the Malware A and B in respective communication process, β when Malware A and B influence each other1 0WithFor empirical value； It is the critical value for describing coupling between Malware B and A, is empirical value, by more
Secondary experiment determines；T indicates particular moment constant when Lagrange multiplier is 0；It indicates
Optimal state variable SA(t)、IA(t)、SB(t)、IB(t)；WithFor arbitrary constant in [0,1], the upper of controlled variable is indicated
Boundary.
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