CN104102823A - Random network construction method - Google Patents

Abstract

The invention discloses a random network construction method, which comprises the main steps: determining a generated network set, calculating an adjacency matrix set for generated networks, determining a degree distribution polynomial set for the generated networks, computing adjacency matrixes of the random networks, calculating the degree distribution of the random networks and the like. According to the random network construction method, a random network is constructed based on a Kronecker sum of the adjacency matrixes of multiple generated networks, and by choosing different generated networks, different numbers of generated networks or adjusting the order of the Kronecker sums of the adjacency matrixes of the generated networks, different random networks can be acquired. The degree distribution of the random network obtained in the method is in normal distribution, and is similar to a classic random network. In addition, by adopting the frequency multiplication and coefficient addition operations of usually polynomial multiplication on the degree distribution polynomial expression form, the degree distribution of the random network can be calculated theoretically strictly. The characteristic that the degree distribution is in normal distribution is substantially originated from a normally-distributed linear combination of the degree distribution of the generated networks.

Description

A kind of random network construction method

Technical field

The invention belongs to electric digital data processing field, be specially adapted to the data processing method of specific function, be specifically related to a kind of based on adjacency matrix Kronecker and random network construction method.

Background technology

The deep rise that promotes Network Science of complex network research, the research object in many fields such as natural science and social science all can be abstracted into complex network research.In the analysis and research that are structured in other fields such as community network, computer network, virtual society network of complex network model, occupy very consequence.The middle of last century, the systematic Study of complex network has been started in the proposition of ER stochastic network model, and becomes the basis of follow-up complex network research nearly half a century; Last century Mo, new era that in succession proposes to have opened up complex network research of WS small-world network model and BA scale-free model.In recent years, self-similarity nature is regarded as the 3rd characteristic of complex network and has received the increasing concern of people, and relevant scholar has proposed LL Self-similar Network model.When every field Network Science is flourish, stochastic network model is still the emphasis of present stage complex network research, and WS small-world network model and NW small-world network model all come from ER stochastic network model.Present stage, the structure of complex network mainly contained following several method:

(1) method based on graph theory

The research of complex network is set something afoot in graph theory, and the research strategy of graph theory has been inherited in the research of complex network.List of references [1] " On the evolution of random graphs " (Erdos P, Renyi A.Publ.Math.Inst.Hung.Acad.Sci., [M] .1960,5:17 – 60) connection between the mode processing node of employing completely random, ER stochastic network model has been proposed---ER (Vertices, Probability), construct the random network (Random Network) of the degree distribution Normal Distribution of node.List of references [2] " Collective dynamics of'small-world'networks " (Watts D J, Strogatz S H.Nature[J] .1998, 393:440 – 442) adopt and reconnect at random the connection between processing node, list of references [3] " Renormalization group analysis of the small-world network model " (Newman M EJ, Watts D J.Phys Lett A[J] .1999, 293:341-346) adopt the connection between random edged processing node, WS network model has been proposed respectively---WS (Vertices, Neighbors, Reprobability) and NW network model---NW (Vertices, Neighbors, AddLink), set forth the Small-world Characters of complex network, construct the small-world network (' Small-world ' Network) of the degree distribution obeys index distribution of node.NW Small World Model is equal to WS Small World Model in essence.List of references [4] " Emergence of Scaling in Random Networks " (Barabasi A L, Albert R.Science[J] .1999,286:509-512) adopt and increase and the preferentially connection between processing node, BA network model has been proposed---BA (InitVertices, InitProbability, AddVertices, AddLink), set forth the scaleless property matter of complex network, the degree that constructs node distributes and obeys the scale-free networks network (Scale-free Network) of power-law distribution.Worldlet characteristic with without characteristics of scale, be described as two large characteristics of complex network, to many researchs of complex network, all based on WS model or BA model, adopt the complex network of these two kinds of methods structures in statistical significance, to there is worldlet or without characteristics of scale.

(2) method based on generating network adjacency matrix

List of references [5] " a kind of construction method of complex network " (Li Tianrui, Liu Shengjiu, pearl is outstanding, Wang Hongjun [P], CN201410092765.2. the .2014-3-13 of Southwest Jiaotong University) the generation series of complex network of the Kronecker product iteration based on a generating network adjacency matrix, construct the Self-similar Network model simultaneously with self similarity and worldlet characteristic---LL (InitVertices, InitProbability, IterNum).Its self-similarity nature comes from by the adjacency matrix of the fractal matrix form of the Kronecker product iteration generation of generating network adjacency matrix, and its worldlet characteristic comes from the twice that its diameter is no more than generating network diameter.Adopt the Self-similar Network degree that the method builds to distribute and can strictly calculate theoretically.

(3) method based on hypergraph or super-network

Limit of common figure can only connect two nodes, and " limit " in hypergraph can comprise a plurality of nodes.List of references [6] " a kind of super-network evolutionary model builds and specificity analysis " (Hu Feng, Zhao Haixing, Ma Xiujuan. Chinese science: physics mechanics uranology [J], 2013,43:16-22) built a kind of super-network Dynamic Evolution Model, analyzed theoretically the characteristic that releases souls from purgatory distribution, and carried out emulation experiment, discovery is along with the increase of network size, model appearance and existing growth and the preferential consistent result of complex network that is connected, several degree of complicated super-network distribute and demonstrate without characteristics of scale.The complex network that adopts the method to build is actually the scale-free networks network of another kind of form.

(4) additive method

Except traditional graph theory, hypergraph and super-network method, additive method is also for the framework of complex network.List of references [7] " the determinacy complex network evolutionary model research based on the fractal pad of Sierpinski " (Xing Changming, Liu Fangai. Acta Physica Sinica [J] .2010,59 (3): 1608-1614) based on the fractal pad of Sierpinski, by the mode of iteration, constructed complex network model and a deterministic unified model S-DUM of small-world network model S-DSWN and two determination increases of scale-free model S-DSFN.List of references [8] " the pyramidal progress of polytype network " (Fang Jinqing, Li Yong, Liu Qiang. complication system and complexity science .2013.10 (2): 69-76) sum up the network pyramid of 3 types, pyramid having summarized network model complicacy pyramid, high-tech network pyramid and broad sense Farey tree tissue, and analyzed these pyramidal feature and character.

Generally speaking, the research of complex network characteristic is still to a large focus of complex network research now, undeniablely be, although random network, small-world network, scale-free networks network and Self-similar Network etc. are all had to comparatively ripe theory and method, most of research also conforms to true complex network, but still the various features of true complex network in the life that cannot comprehensively reflect reality, need further every characteristic of further investigation complex network, to the research of the random network as basic, should cause enough attention.Wherein, the structure of network model is the most important thing.

Summary of the invention

In order to overcome the above-mentioned shortcoming of prior art, the invention provides a kind of random network construction method, improve the construction method of existing complex network, make it to possess computation complexity low, the feature easily realizing.The technical solution adopted for the present invention to solve the technical problems is: a kind of new random network construction method, by determining generating network set, calculate the set of generating network adjacency matrix, determine the set of generating network degree distribution polynomial expression, calculate random network adjacency matrix, calculate the steps such as degree distribution of random network, the Kronecker based on a plurality of generating network adjacency matrix and obtain random network.By choose the generating network of different generating networks, varying number or adjust generating network adjacency matrix Kronecker and order can obtain different random networks; Comprise the steps:

(1) determine generating network set U _g={ G ₁, G ₂, G ₃..., G _i...;

(2) calculate generating network set U _gmiddle all-network G _iadjacency matrix A (G _i), obtain generating network set U _gadjacency matrix set U _{a (G)}={ A (G ₁), A (G ₂), A (G ₃) ..., A (G _i) ... }:

At generating network set U _gin, for arbitrary generating network G with n node, its adjacency matrix A (G) is the square formation of n * n, wherein for each data in square formation, if node i is adjacent with node j, there is A (G) (i, j)=1, otherwise, A (G) (i, j)=0; If the number of links of generating network G is m, in adjacency matrix A (G), 1 number is also m, and the network density of generating network

(3) calculate generating network set U _gmiddle all-network G _idegree distribute, obtain generating network set U _gdegree distribution polynomial expression set U _{poly (G)}={ Poly (G ₁), Poly (G ₂), Poly (G ₃) ..., Poly (G _i) ... }:

At generating network set U _gin, for arbitrary generating network G with n node, its degree distribution polynomial expression form Poly (G) can be expressed as follows:

$\mathrm{Poly}\left(G\right)=\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{x}^{D_i}=\underset{j=1}{\overset{\∞}{\mathrm{\Σ}}}{N}_j{x}^j---\left(1\right)$

Wherein, n is interstitial content, D _ithe degree that represents i node, N _jdegree of a representation is the number of the node of j;

(4) from generating network set, choose in turn k generating network G ₍₁₎, G ₍₂₎, G ₍₃₎..., G _(k-1), G _(k), be designated as G ₍₁₎g ₍₂₎g ₍₃₎g _(k-1)g _(k), allow to repeat to choose, to each generating network G _(i)corresponding adjacency matrix A (G _(i)) calculate as follows the adjacency matrix A of constructed random network ^(l)(G ^(l)), wherein, l represents the number of times of computing, A ^(l)(G ^(l)) represent the adjacency matrix of the new random network that l adjacency matrix corresponding to generating network carries out obtaining after computing in turn:

According to Kronecker and rule carry out computing, obtain the adjacency matrix of constructed new random network; Matrix A (a _ij) _{m * m}and matrix B (b _ij) _{n * n}kronecker and be defined as follows:

$A_{m\×m}\⊕B_{n\×n}=A_{m\×m}\⊗I_{n\×n}+I_{m\×m}\⊗B_{n\×n}---\left(2\right)$

I wherein _{n * n}represent n * n unit matrix, represent Kronecker product calculation, matrix P _{p * p}with matrix Q _{q * q}kronecker product be defined as follows:

Write for convenience, adopt the method to choose in turn k generating network G ₍₁₎, G ₍₂₎, G ₍₃₎..., G _(k-1), G _(k)and the random network G obtaining ^(k)can be designated as G ^(k)=G ₍₁₎g ₍₂₎g ₍₃₎g _(k-1)g _(k).

(5) calculate as follows the degree distribution PolyDD (G of constructed new random network ^(l)), wherein, l represents the number of times of computing, PolyDD (G ^(l)) represent that l adjacency matrix corresponding to generating network carries out the new random network degree distribution polynomial expression obtaining after computing in turn:

$\begin{array}{c}\mathrm{PolyDD}\left(G^{(k+1)}\right)=\mathrm{DD}(\mathrm{Poly}\left(G^{\left(k\right)}\right),\mathrm{Poly}\left(G_{(k+1)}\right))\\ =\mathrm{DD}(\underset{i=1}{\overset{\∞}{\mathrm{\Σ}}}{N}_i^{\left(k\right)}{x}^i,\underset{j=1}{\overset{\∞}{\mathrm{\Σ}}}{N}_{(k+1)j}{x}^j)\\ \underset{i=1}{\overset{\∞}{\mathrm{\Σ}}}\underset{j=1}{\overset{\∞}{\mathrm{\Σ}}}{N}_i^{\left(k\right)}{N}_{(k+1)j}{x}^{i+j}(i,j=\mathrm{1,2,3},...)\end{array}---\left(4\right)$

(6) repeating step (4) and step (5), when obtaining specified node number, given link number or specifying the random network of generating network number, terminating operation.

Compared with prior art, good effect of the present invention is:

One, be different from classical complex network and mainly by adding node, build complex network, by adjacency matrix, build complex network, improved the construction method of existing complex network, computation complexity is low, easily realizes.

Classical complex network construction method is all that the connection by initial base net network is added between node of different nature or knot modification obtains, and the research that builds complex network from the adjacency matrix of generating network is less.Compared to other complex network construction methods, the present invention is based on the Kronecker and the structure of realizing stochastic network model of a plurality of generating network adjacency matrix, only relate to matrix operation.Because matrix operation is the basis of existing most of Software tool (as MATLAB etc.), the complex network construction method that the present invention proposes is easy to realize by existing instrument.

Two, being different from Self-similar Network model is that Kronecker based on a generating network adjacency matrix amasss to build complex network, the present invention is the Kronecker of the generating network adjacency matrix based on a plurality of variety classeses, varying number, different order and builds random network, expanded its range of application.

Present stage, existing self similarity complex network model was based on a definite generating network, Kronecker product by its adjacency matrix builds complex network iteratively, can only generate the complex network that particular sections is counted out or particular sections is counted out, range of application is comparatively narrow.The present invention is based on a set that comprises a plurality of generating networks and build complex network, by generating network kind, generating network quantity, generating network are sequentially adjusted, and adjacency matrix corresponding to generating network carried out to Kronecker and computing can obtain different complex networks, greatly expanded its range of application.

Three, the degree that is different from classical complex network model distributes and to obtain by statistical method, the degree distribution polynomial expression form of Adoption Network, distributes by the degree that can strictly calculate theoretically this type of random network to its computing that degree distribution polynomial expression form adopts the number of times of common polynomial multiplication to multiply each other and coefficient is added.Adopt the degree of the random network that the present invention generates to distribute to be the characteristic of normal distribution to come from fact the linear combination of generating network degree distribution normal distribution.

The worldlet characteristic of complex network and all obtaining by statistical method without characteristics of scale, the degree of directly adding up especially complex network without characteristics of scale distributes and obtains, but degree is distributed and lacks directly perceived, vivid understanding and understanding.Be similar in chemosynthesis by little molecular chemistry material synthetic macromolecule chemical substance, adopt the resulting random network of the present invention, its degree distributes and can by the multiplication of common polynomial multiplication and the method for number of times addition, strictly calculate theoretically by existing instrument.Research shows, the random network degree distribution Normal Distribution that adopts this kind of method to obtain, with classical stochastic network model degree distributional class seemingly.

Accompanying drawing explanation

Examples of the present invention will be described by way of reference to the accompanying drawings, wherein:

Fig. 1 adopts stochastic network model ER (10000, the 0.5) degree that document [1] generates to distribute;

Fig. 2 adopts small-world network model WS (10000,100, the 0.5) degree that document [2] generates to distribute;

Fig. 3 adopts scale-free model BA (10,0.5,10000, the 5) degree that document [4] generates to distribute;

Fig. 4 adopts Self-similar Network model LL (6,0.4, the 25) degree that document [5] generates to distribute;

Fig. 5 is U _gin 6 generating network G ₁, G ₂, G ₃, G ₄, G ₅, G ₆topology diagram;

Fig. 6 is U _{a (G)}in 6 generating network G ₁, G ₂, G ₃, G ₄, G ₅, G ₆adjacency matrix;

Fig. 7 adopts the random new complex network G generating of the present invention ⁽²⁴⁾degree distributes, wherein:

G ⁽²⁴⁾＝G ₅G ₂G ₅G ₅G ₁G ₄G ₅G ₂G ₆G ₄G ₁G ₅G ₁G ₁G ₄G ₅G ₆G ₂G ₁G ₅G ₄G ₁G ₄G ₃

Fig. 8 is the new complex network G that adopts the present invention to generate ⁽²⁴⁾degree distributes, wherein:

G ⁽²⁴⁾＝G ₃G ₄G ₅G ₆G ₃G ₄G ₅G ₆G ₃G ₄G ₅G ₆G ₃G ₄G ₅G ₆G ₃G ₄G ₅G ₆G ₃G ₄G ₅G ₆；

Fig. 9 is the new complex network G that adopts the present invention to generate ⁽²⁴⁾degree distributes, wherein:

G ⁽²⁴⁾＝G ₁G ₂G ₃G ₄G ₅G ₆G ₆G ₅G ₄G ₃G ₂G ₁G ₁G ₂G ₃G ₄G ₅G ₆G ₆G ₅G ₄G ₃G ₂G ₁；

Figure 10 is the new complex network G that adopts the present invention to generate ⁽²⁴⁾degree distributes, and 24 generating networks wherein all generate at random.

Figure 11 is U _{poly (G)}in 6 generating network G ₁, G ₂, G ₃, G ₄, G ₅, G ₆degree distribution polynomial expression;

embodiment

A new random network construction method, comprises the steps:

(1) determine generating network set U _g={ G ₁, G ₂, G ₃..., G _i...;

(2) calculate generating network set U _gmiddle all-network G _iadjacency matrix A (G _i), obtain generating network set U _gadjacency matrix set U _{a (G)}={ A (G ₁), A (G ₂), A (G ₃) ..., A (G _i) ... }:

At generating network set U _gin, for arbitrary generating network G with n node, its adjacency matrix A (G) is the square formation of n * n, wherein for each data in square formation, if node i is adjacent with node j, there is A (G) (i, j)=1, otherwise, A (G) (i, j)=0; If the number of links of generating network G is m, in adjacency matrix A (G), 1 number is also m, and the network density of generating network

(3) calculate generating network set U _gmiddle all-network G _idegree distribute, obtain generating network set U _gdegree distribution polynomial expression set U _{poly (G)}={ Poly (G ₁), Poly (G ₂), Poly (G ₃) ..., Poly (G _i) ... }:

At generating network set U _gin, for arbitrary generating network G with n node, its degree distribution polynomial expression form Poly (G) can be expressed as follows:

$\mathrm{Poly}\left(G\right)=\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{x}^{D_i}=\underset{j=1}{\overset{\∞}{\mathrm{\Σ}}}{N}_j{x}^j---\left(1\right)$

Wherein, n is interstitial content, D _ithe degree that represents i node, N _jdegree of a representation is the number of the node of j;

(4) from generating network set, choose in turn k generating network G ₍₁₎, G ₍₂₎, G ₍₃₎..., G _(k-1), G _(k), be designated as G ₍₁₎g ₍₂₎g ₍₃₎g _(k-1)g _(k), allow to repeat to choose, to each generating network G _(i)corresponding adjacency matrix A (G _(i)) calculate as follows the adjacency matrix A of constructed random network ^(l)(G ^(l)), wherein, l represents the number of times of computing, A ^(l)(G ^(l)) represent the adjacency matrix of the new random network that l adjacency matrix corresponding to generating network carries out obtaining after computing in turn:

According to Kronecker and rule carry out computing, obtain the adjacency matrix of constructed new random network; Matrix A (a _ij) _{m * m}and matrix B (b _ij) _{n * n}kronecker and be defined as follows:

$A_{m\×m}\⊕B_{n\×n}=A_{m\×m}\⊗I_{n\×n}+I_{m\×m}\⊗B_{n\×n}---\left(2\right)$

I wherein _{n * n}represent n * n unit matrix, represent Kronecker product calculation, matrix P _{p * p}with matrix Q _{q * q}kronecker product be defined as follows:

Write for convenience, adopt the method to choose in turn k generating network G ₍₁₎, G ₍₂₎, G ₍₃₎..., G _(k-1), G _(k)and the random network G obtaining ^(k)can be designated as G ^(k)=G ₍₁₎g ₍₂₎g ₍₃₎g _(k-1)g _(k).

(5) calculate as follows the degree distribution PolyDD (G of constructed new random network ^(l)), wherein, l represents the number of times of computing, PolyDD (G ^(l)) represent that l adjacency matrix corresponding to generating network carries out the new random network degree distribution polynomial expression obtaining after computing in turn:

$\begin{array}{c}\mathrm{PolyDD}\left(G^{(k+1)}\right)=\mathrm{DD}(\mathrm{Poly}\left(G^{\left(k\right)}\right),\mathrm{Poly}\left(G_{(k+1)}\right))\\ =\mathrm{DD}(\underset{i=1}{\overset{\∞}{\mathrm{\Σ}}}{N}_i^{\left(k\right)}{x}^i,\underset{j=1}{\overset{\∞}{\mathrm{\Σ}}}{N}_{(k+1)j}{x}^j)\\ \underset{i=1}{\overset{\∞}{\mathrm{\Σ}}}\underset{j=1}{\overset{\∞}{\mathrm{\Σ}}}{N}_i^{\left(k\right)}{N}_{(k+1)j}{x}^{i+j}(i,j=\mathrm{1,2,3},...)\end{array}---\left(4\right)$

(6) repeating step (4) and step (5), when obtaining specified node number, given link number or specifying the random network of generating network number, terminating operation.

For choosing in turn k generating network G ₍₁₎, G ₍₂₎, G ₍₃₎..., G _(k-1), G _(k)and the random network G obtaining ^(k), the nodes of the random network obtaining number of links network density $\mathrm{Density}\left(k\right)=\frac{2\underset{i=1}{\overset{k}{\mathrm{\Π}}}m\left(G_{\left(i\right)}\right)}{\underset{i=1}{\overset{k}{\mathrm{\Σ}}}\left[m\left(G_{\left(i\right)}\right)\underset{\underset{j\≠1}{j=1}}{\overset{j=k}{\mathrm{\Π}}}n\left(G_{\left(j\right)}\right)\right]\left\{\underset{i=1}{\overset{k}{\mathrm{\Σ}}}\right[m\left(G_{\left(i\right)}\right)\underset{\underset{j\≠1}{j=1}}{\overset{j=k}{\mathrm{\Π}}}n\left(G_{\left(j\right)}\right)]-1\}},$ N (G wherein _(i)) representing the nodes of generating network, m (G (i)) represents the number of links of generating network.

The new random network construction method that adopts the present invention to propose, based on a set that comprises a plurality of generating networks, Kronecker and the computing of the adjacency matrix by a series of generating networks of choosing in turn build random network, and its degree distributes and is normal distribution.By adjusting generating network kind, generating network quantity and generating network order, can obtain different random networks, similar with classical stochastic network model, be different from classical small-world network model and scale-free model, be also different from the Self-similar Network model building based on a generating network.The degree that can strictly calculate theoretically this type of random network by the computing that adopts the number of times of common polynomial multiplication to multiply each other to generating network degree distribution polynomial expression form and coefficient is added distributes.

Emulation experiment

In order to verify that the present invention is based on one comprises a plurality of generating network set, and by the corresponding Kronecker of adjacency matrix and the validity of the random network construction method obtaining of generating network of choosing in turn, carried out emulation experiment.

Determine the set U of a generating network _g, the topology diagram of 6 generating networks wherein as shown in Figure 5, the adjacency matrix set U that these 6 generating networks are corresponding _{a (G)}as shown in Figure 6, corresponding degree distribution polynomial expression form set U _{poly (G)}as shown in figure 11.

To figure G ₁, figure G ₁have 4 nodes, wherein, the degree of the 1st node is that the degree of 3, the 2 nodes is that the degree of 2, the 3 nodes is that the degree of 2, the 4 nodes is 1, therefore its degree distribution polynomial expression form is: Poly (G)=x+2x ²+ x ³.

To figure G ₂, figure G ₂have 4 nodes, wherein, the degree of the 1st node is that the degree of 3, the 2 nodes is that the degree of 3, the 3 nodes is that the degree of 2, the 4 nodes is 2, therefore its degree distribution polynomial expression form is: Poly (G)=2x ²+ 2x ³.

To figure G ₃, figure G ₃have 5 nodes, wherein, the degree of the 1st node is that the degree of 3, the 2 nodes is that the degree of 1, the 3 node is that the degree of 2, the 4 nodes is that the degree of 2, the 5 nodes is 2, therefore its degree distribution polynomial expression form is: Poly (G)=x+3x ²+ x ³.

To figure G ₄, figure G ₄have 5 nodes, wherein, the degree of the 1st node is that the degree of 2, the 2 nodes is that the degree of 3, the 3 nodes is that the degree of 3, the 4 nodes is that the degree of 1, the 5 node is 1, therefore its degree distribution polynomial expression form is: Poly (G)=2x+x ²+ 2x ³.

To figure G ₅, figure G ₅have 6 nodes, wherein, the degree of the 1st node is 3, the degree of the 2nd node is that the degree of 1, the 3 node is that the degree of 2, the 4 nodes is 2, the degree of the 5th node is that the degree of 3, the 6 nodes is 1, therefore its degree distribution polynomial expression form is: Poly (G)=2x+2x ²+ 2x ³.

To figure G ₆, figure G ₆have 6 nodes, wherein, the degree of the 1st node is 3, the degree of the 2nd node is that the degree of 2, the 3 nodes is that the degree of 2, the 4 nodes is 2, the degree of the 5th node is that the degree of 2, the 6 nodes is 1, therefore its degree distribution polynomial expression form is: Poly (G)=x+4x ²+ x ³.

Experiment one

At generating network set U _gin choose at random 24 generating networks, allow to repeat, adopt the present invention to build the complex network obtaining and be designated as G ⁽²⁴⁾, wherein:

G ⁽²⁴⁾＝G ₅G ₂G ₅G ₅G ₁G ₄G ₅G ₂G ₆G ₄G ₁G ₅G ₁G ₁G ₄G ₅G ₆G ₂G ₁G ₅G ₄G ₁G ₄G ₃

Its degree distributes as shown in Figure 7.

Experiment two

At generating network set U _gmiddle circulation is chosen generating network G 6 times in turn ₃, G ₄, G ₅, G ₆, adopt the present invention to build the complex network obtaining and be designated as G ⁽²⁴⁾, wherein:

G ⁽²⁴⁾＝G ₃G ₄G ₅G ₆G ₃G ₄G ₅G ₆G ₃G ₄G ₅G ₆G ₃G ₄G ₅G ₆G ₃G ₄G ₅G ₆G ₃G ₄G ₅G ₆；

Its degree distributes as shown in Figure 8.

Experiment three

At generating network set U _g2 times positive sequences of middle circulation, inverted order are chosen generating network G in turn ₁, G ₂, G ₃, G ₄, G ₅, G ₆, adopt the present invention to build the complex network obtaining and be designated as G ⁽²⁴⁾, wherein:

G ⁽²⁴⁾＝G ₁G ₂G ₃G ₄G ₅G ₆G ₆G ₅G ₄G ₃G ₂G ₁G ₁G ₂G ₃G ₄G ₅G ₆G ₆G ₅G ₄G ₃G ₂G ₁；

Its degree distributes as shown in Figure 9.

Experiment four

24 generating networks of random generation, adopt the present invention to build the complex network obtaining and are designated as G ⁽²⁴⁾,

Its degree distributes as shown in figure 10.

From Fig. 7, Fig. 8, Fig. 9, Figure 10 can find out, adopt the random network degree distribution that invention obtains to be normal distribution, to classical ER stochastic network model similar (as Fig. 1), be different from WS small-world network model (as Fig. 2 etc.), BA scale-free model (as Fig. 3 etc.) and LL Self-similar Network model similar (as Fig. 4 etc.), main cause be the Kronecker based on a plurality of generating network adjacency matrix that the present invention proposes and and the degree that obtains complex network distributes and is equivalent to the distribute linear combination of a plurality of separate normal distributions of generating network degree, its degree distributes and is still normal distribution.From Fig. 7, Fig. 8, Fig. 9, Figure 10, it can also be seen that the random network that adopts the present invention to obtain can obtain different random networks by adjusting generating network kind, generating network quantity and generating network order.Comparison diagram 1 can find out with Fig. 7, Fig. 8, Fig. 9, Figure 10, and distribute degree that more classical ER stochastic network model generates random network of the degree that adopts the present invention to generate random network distributes more smoothly, can reach quickly steady state (SS).

Claims (3)

1. a random network construction method, by determining generating network set, calculate the set of generating network adjacency matrix, determine the set of generating network degree distribution polynomial expression, calculate the adjacency matrix of random network, calculate the degree of random network and distribute, the Kronecker based on a plurality of generating network adjacency matrix and obtain random network; By choose the generating network of different generating networks, varying number or adjust generating network adjacency matrix Kronecker and order to obtain different random networks: comprise the steps:

(1) determine generating network set U _g={ G ₁, G ₂, G ₃..., G _i...;

(2) calculate generating network set U _gmiddle all-network G _iadjacency matrix A (G _i), obtain generating network set U _gadjacency matrix set U _{a (G)}={ A (G ₁), A (G ₂), A (G ₃) ..., A (G _i) ... }:

At generating network set U _gin, for arbitrary generating network G with n node, its adjacency matrix A (G) is the square formation of n * n, wherein for each data in square formation, if node i is adjacent with node j, there is A (G) (i, j)=1, otherwise, A (G) (i, j)=0; If the number of links of generating network G is m, in adjacency matrix A (G), 1 number is also m, and the network density of generating network

(3) calculate generating network set U _gmiddle all-network G _idegree distribute, obtain the degree distribution polynomial expression set U of generating network set UG _{poly (G)}={ Poly (G ₁), Poly (G ₂), Poly (G ₃) ..., Poly (G _i) ... }:

At generating network set U _gin, for arbitrary generating network G with n node, its degree distribution polynomial expression form Poly (G) can be expressed as follows:

$\mathrm{Poly}\left(G\right)=\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{x}^{D_i}=\underset{j=1}{\overset{\∞}{\mathrm{\Σ}}}{N}_j{x}^j---\left(1\right)$

Wherein, n is interstitial content, D _ithe degree that represents i node, N _jdegree of a representation is the number of the node of j;

(4) from generating network set, choose in turn k generating network G ₍₁₎, G ₍₂₎, G ₍₃₎..., G _(k-1), G _(k), be designated as G ₍₁₎g ₍₂₎g ₍₃₎g _(k-1)g _(k), allow to repeat to choose, to each generating network G _(i)corresponding adjacency matrix A (G _(i)) calculate as follows the adjacency matrix A of constructed random network ^(l)(G ^(l)), wherein, l represents the number of times of computing, A ^(l)(G ^(l)) represent the adjacency matrix of the new random network that l adjacency matrix corresponding to generating network carries out obtaining after computing in turn:

According to Kronecker and rule carry out computing, obtain the adjacency matrix of constructed new random network; Matrix A (a _ij) _{m * m}and matrix B (b _ij) _{n * n}kronecker and be defined as follows:

$A_{m\×m}\⊕B_{n\×n}=A_{m\×m}\⊗I_{n\×n}+I_{m\×m}\⊗B_{n\×n}---\left(2\right)$

I wherein _{n * n}represent n * n unit matrix, represent Kronecker product calculation, matrix P _{p * p}with matrix Q _{q * q}kronecker product be defined as follows:

Write for convenience, adopt the method to choose in turn k generating network G ₍₁₎, G ₍₂₎, G ₍₃₎..., G _{(k -1)}, G _(k)and the random network G obtaining ^(k)can be designated as G ^(k)=G ₍₁₎g ₍₂₎g ₍₃₎g _(k-1)g _(k).

(5) calculate as follows the degree distribution PolyDD (G of constructed new random network ^(l)), wherein, l represents the number of times of computing, PolyDD (G ^(l)) represent that l adjacency matrix corresponding to generating network carries out the new random network degree distribution polynomial expression obtaining after computing in turn:

$\begin{array}{c}\mathrm{PolyDD}\left(G^{(k+1)}\right)=\mathrm{DD}(\mathrm{Poly}\left(G^{\left(k\right)}\right),\mathrm{Poly}\left(G_{(k+1)}\right))\\ =\mathrm{DD}(\underset{i=1}{\overset{\∞}{\mathrm{\Σ}}}{N}_i^{\left(k\right)}{x}^i,\underset{j=1}{\overset{\∞}{\mathrm{\Σ}}}{N}_{(k+1)j}{x}^j)\\ \underset{i=1}{\overset{\∞}{\mathrm{\Σ}}}\underset{j=1}{\overset{\∞}{\mathrm{\Σ}}}{N}_i^{\left(k\right)}{N}_{(k+1)j}{x}^{i+j}(i,j=\mathrm{1,2,3},...)\end{array}---\left(4\right)$

(6) repeating step (4) and step (5), when obtaining specified node number, given link number or specifying the random network of generating network number, terminating operation.

2. a kind of random network construction method according to claim 1, is characterized in that: determine generating network set U _gtime, select interstitial content n be less than or equal to 10 and node between connect less simple network as generating network.

3. a kind of new random network construction method according to claim 1, is characterized in that: for choosing in turn k generating network G ₍₁₎, G ₍₂₎, G ₍₃₎..., G _(k-1), G _(k)and the random network G obtaining ^(k), the nodes of the random network obtaining $n\left(k\right)=\underset{i=1}{\overset{k}{\mathrm{\Π}}}n\left(G_{\left(i\right)}\right),$ Number of links $n\left(k\right)=\underset{i=1}{\overset{k}{\mathrm{\Σ}}}\left[m\left(G_{\left(i\right)}\right)\underset{\underset{j\≠1}{j=1}}{\overset{j=k}{\mathrm{\Π}}}n\left(G_{\left(j\right)}\right)\right],$ Network density $\mathrm{Density}\left(k\right)=\frac{2\underset{i=1}{\overset{k}{\mathrm{\Π}}}m\left(G_{\left(i\right)}\right)}{\underset{i=1}{\overset{k}{\mathrm{\Σ}}}\left[m\left(G_{\left(i\right)}\right)\underset{\underset{j\≠1}{j=1}}{\overset{j=k}{\mathrm{\Π}}}n\left(G_{\left(j\right)}\right)\right]\left\{\underset{i=1}{\overset{k}{\mathrm{\Σ}}}\right[m\left(G_{\left(i\right)}\right)\underset{\underset{j\≠1}{j=1}}{\overset{j=k}{\mathrm{\Π}}}n\left(G_{\left(j\right)}\right)]-1\}},$ N (G wherein _(i)) represent the nodes of generating network, m (G _(i)) represent the number of links of generating network.