CN106934422A - Based on the level vision abstract method for improving power derivation graph layout - Google Patents

Abstract

A kind of level vision abstract method based on improvement power derivation graph layout, step is as follows：The first step, using improved power guiding algorithm, has in generation one substantially cluster structure, can embody the preliminary placement of graph structure information；Second step, based on layout result, the hierarchical structure of figure is generated using hierarchy clustering method from bottom to top；3rd step, according to the hierarchical structure of generation, carries out vision abstract in different levels.

Description

Based on the level vision abstract method for improving power derivation graph layout

Technical field

The present invention relates to the graphic display method of network data.

Background technology

Network data refers to the data that can be expressed as graph structure, complicated relation between expression things.From various societies Understand relational network to scientific collaboration network, biological contexts network to computer network etc., human lives are full of various networks The world in.And many networks have the feature in the small-sized world, i.e. these networks have the structure of Local Clustering, they Connectivity high causes that finding a suitable layout and graph structure is difficult to.Node link method is the expression directly perceived of network, with top Put to represent the individuality in network, relation individual and between individuality is represented with side.Things in real world is abstracted into section Point link structure, can effective visual network structure by figure placement algorithm.

Artwork is intended to be a physical system by the figure placement algorithm of power guiding, and node regards steel loop as, while regard spring as, system After being endowed certain original state, the effect of power can cause that steel loop is moved, by successive ignition until finding the position of all nodes Put so that stop when gross energy in this system is minimum, at this moment just reach the dynamic equilibrium of whole layout.It is easy to reason Solution, realizes simply, to be applicable most of Network data sets, and the algorithm realizes that effect has preferable symmetry, than Relatively meet Aesthetic Standards.Also, it can entirely be laid out the process for gradually tending to convergence stabilization with Dynamic Display, so that user is to cloth Office's result is easier to receive and understands.Power guiding algorithm is proposed then have many scholars to attempt introducing at first by Peter Eades Different mechanical models creates more attractive layout, existing comparing it is famous have FR algorithms, DH algorithms and KK algorithms.But It is that the expansion of network data scale causes serious juxtaposition phenomenon, hinders cognition of the user to True Data, it is difficult to examine Survey graphic structure, the contact explored between individuality.Additionally, for the relational network with small-sized world feature, it to be represented Information out is also very limited, or even probably destroys those with the original for disclosing relation and function between network structure and function The cluster that begins structure.LinLog power guiding algorithm connects the degree of coupling between nodal distance and node, is embodied with distance Just, resulting layout result can clearly display the cluster in figure to the degree of coupling.But LinLog algorithms are applied to It is in real network data test result indicate that, since a random original state, it is more easily trapped into than conventional method Local minimum.

The content of the invention

The present invention will overcome the disadvantages mentioned above of prior art, propose it is a kind of based on improve the expansible of power guiding layout can Depending on changing level abstract method, to solve the figure location problem of the complex network with small-sized world feature on a large scale, figure is kept Original structure, while reducing potential VC.Both user can have been allowed to see the general view of figure, had been had to different data distributions Individual overall understanding, it is also possible to finer observation is carried out to local cluster, help finds special point or cluster structure.

The present invention is improved with the advantage of LinLog algorithms with reference to FR algorithms to power guiding algorithm, generates not only attractive in appearance but also energy Keep the preliminary placement of figure cluster structure；Layout result is then based on, using hierarchy clustering method generation figure from bottom to top Hierarchical structure, shows determining the cluster under different levels, it is allowed to which user is more while defining and embodying the parameter of abstract level Individual level observed data architectural feature.

Technical scheme is altogether in three steps：The first step, using improved power guiding algorithm, generation one has Substantially cluster structure, the preliminary placement of graph structure information can be embodied；Second step, based on layout result, using from bottom to top Hierarchy clustering method generates the hierarchical structure of figure；3rd step, according to the hierarchical structure of generation, carries out vision and takes out in different levels As.

A kind of level vision abstract method based on improvement power derivation graph layout, step is as follows：

1. preliminary placement is generated using improved power guiding algorithm.Improved power guiding algorithm be exactly first with FR algorithms from Beginning is put in seat in the plane, and iterate searching state of minimum energy, produces a relatively stable layout, then perform based on the state LinLog algorithms.

1.1 initialization：Primary condition is set, each node v is assigned_iOne random site p_i；

1.2 figures based on FR algorithms are laid out

To each node v in figure_iMake following iteration：

What 1.2.1 calculate node was subject to makes a concerted effort：A) first calculate node v_iWith other nodes v_jBetween Euclidean distance x=| p_i-p_j|.B) substitution formula (1) and (2) calculates the gravitation and repulsion from other nodes suffered by the node respectively.C) according to public affairs Formula (3) calculate node v_iSuffered F (the v that make a concerted effort_i)

f(e_ij)=x²/k (1)

g(v_i, v_j)=k²/x (2)

Wherein, k=sqrt (A/ | V |), A are the viewing area area of whole view, and | V | is node total number；e_ijRepresent section Point v_iWith v_jBetween the side that is connected, therefore formula (1) calculate two have the connected node in side between gravitation.Formula (2) calculates at 2 points Between repulsion.In formula (3)Represent the direction of power.

1.2.2 node location updates：Node v_iAccording to the effect F (v that make a concerted effort above_i) move, reach new position p_i'=p_i+C*F(v_i).Here C is common constant, is the influence coefficient of power.When C is larger, joints influence is big, motion Faster；Vice versa.

1.2.3 iteration termination condition is judged：The gross energy E of whole system, then calculate node iteration are calculated according to formula (4) Energy differences Δ E before and after location updating, when Δ E is continuously less than the threshold value that certain sets, then it is assumed that system reaches stable state, Stop iteration.

Figure layout under 1.3LinLog algorithms

Based on the layout of FR algorithms, the position of each node is recorded, to each node v in figure_iMake following iteration：

What 1.3.1 calculate node was subject to makes a concerted effort：A) first calculate node v_iWith other nodes v_jBetween Euclidean distance x=| p_i-p_j|.B) substitution formula (5) and (6) calculates the gravitation and repulsion from other nodes suffered by the node respectively.C) according to public affairs Formula (3) calculate node v_iSuffered F (the v that make a concerted effort_i)

f(e_ij)=C₁ (5)

g(v_i, v_j)=C₂/x (6)

C in formula₁And C₂All it is constant.

1.3.2 node location updates：Node v_iAccording to the effect F (v that make a concerted effort above_i) move, reach new position p_i′。

1.3.3 iteration termination condition is judged：The gross energy E of whole system, then calculate node iteration are calculated according to formula (7) Energy differences Δ E before and after location updating, when Δ E is continuously less than the threshold value that certain sets, then it is assumed that system reaches stable state, Stop iteration.

2. the layout result based on power guiding algorithm, generates the hierarchical structure of figure.

2.1 definition cluster c are five-tuple (l_c,r_c,w_c,f_c,V_c), wherein l_cAnd r_cTwo son clusters of cluster are represented respectively In left subgroup and right subgroup, w_cIts weights are represented, as (distance can select different degree to the distance between its sub- cluster Amount), f_cIt is his father's cluster, V_cThen represent original node set in the figure included in the cluster.

All ancestor node v in 2.2 initialization figures_iFor leaf clusters c_i(u,u,0,u,v_i), u is expressed as sky here, while The position Pc of initialization leaf cluster_iIt is the v after exertin guiding layout_iPosition p_i。

2.3 calculate the distance between cluster.According to the different abstract needs of network data and vision, can select it is different away from Calculated from measurement.

2.3.1 the Euclidean distance r between cluster is calculated_cicj=| Pc_i-Pc_j| as distance metric.

2.3.2 the average path length l between cluster is calculated_cicjAs distance metric.

2.3.3 according to formula (8) by the average path length l between cluster_cicjWith the Betweenness Centrality C of cluster_B(c_i) Sum is combined, used as distance metric.

Wherein α ∈ (0,1) are weighting parameter, represent the shared weight in weights of Betweenness Centrality.α is bigger, then intermediary Influence of the centrality to weights is bigger.Ω_ijIt is node v_iAnd v_jBetween shortest path bar number, Ω_ij(v_k) it is section in these paths Point v_kThe number of times of appearance, therefore, C_B(v_k) represent node v_kBetweenness Centrality.Cluster c_kBetweenness Centrality C_B(c_k) poly- for this The average value of the node Betweenness Centrality that class is included.

2.4 select two cluster c_iWith c_jApart from d_ijRecently, the new cluster c of generation is merged as left and right subgroup_k (c_i,c_j,d_ij,u,(V_ci∪V_cj)), position Pc_kThen according to the node set Vc that new cluster is included_kThe mean place of interior joint To determine.Meanwhile, newly cluster c_kAs c_iWith c_jFather cluster, c_iWith c_j(u, u, 0, c are updated to respectively_k,V_ci) and (u, u, 0, c_k,V_cj), and by c_kAdd and calculate.

2.5 repeat step 2.3 and 2.4 always, find closest cluster and merge repeatedly, until all clusters all Untill merging into a cluster.

3. it is abstract vision to be carried out in different levels.

3.1 will finally merge the cluster c for obtaining when building hierarchical structure_rootIt is defined as root cluster

3.2 define abstract level parameter AL ∈ [0,1], with AL × w_crootHierarchical structure is intercepted to spend, only will symbol Conjunction condition w_ck≤AL×w_croot<w_fcCluster be shown on view, complete it is abstract to the vision of network structure.By abstract Level AL's changes to obtain different abstract results.

The present invention describes a kind of level vision abstract method based on improvement power derivation graph layout, by the net of user input Network data are converted into node link figure, then carry out figure layout using improved power guiding algorithm, obtain that graph structure can be embodied The preliminary placement result of information, it is ensured that the layout structure of figure visually overall aesthetic；Layout result is next based on, using several The measurement that what distance, shortest path, shortest path add 3 kinds of Betweenness Centrality different carries out hierarchy clustering method life from bottom to top Into the hierarchical structure of figure, hierarchy clustering method iterates to calculate the similitude between cluster, similitude cluster high is merged.It is logical Cross this merging, while also reducing the quantity on side, can reach taking out for figure by a similar class node aggregation under clustering measure As simplification, decreased while highlighting global structure by data scale becomes the VC for being brought greatly；Additionally, defining Embody the parameter of abstract level and show determining the cluster under different levels, different level of abstractions are drawn according to abstract level parameter Under figure layout result.When abstract level is high, graph abstraction helps stressing main structure, alleviates and perceives burden；Abstract level is low When, the CONSTRUCTED SPECIFICATION of figure can be observed.Therefore this method can be provided from multiple abstraction levels and remove comprehensive understanding graphic structure.

Advantages of the present invention is as follows：

(1) realize simple.The figure layout method used in the present invention is entered on the basis of traditional power guiding placement algorithm Row is improved, and is realized fairly simple.

(2) scalability is good.Hierarchical structure construction method of the invention, can generate not according to different distance metrics Same hierarchical structure, so as to obtain different abstract results, the different structure attribute of protrusion and feature.

(3) applicability is wide.The level vision abstract method that the present invention is realized is applied to the figure of different scales and type, gathers The quantity of class is generated also without previously given by the regulation of parameter in different levels, meets the multiple need of user Will.

(4) multi-angle observation, multi-layer is presented.The level vision abstract method realized using the present invention, user can be Different level observed data, can capture overall structure change, and the minutia of data can be probed into again, and help excavates one A little hiding structural informations.

Brief description of the drawings

Fig. 1 is flow chart of the invention

Specific embodiment

Technical scheme is altogether in three steps：The first step, using improved power guiding algorithm, generation one has Substantially cluster structure, the preliminary placement of graph structure information can be embodied；Second step, based on layout result, using from bottom to top Hierarchy clustering method generates the hierarchical structure of figure；3rd step, according to the hierarchical structure of generation, carries out vision and takes out in different levels As.

A kind of level vision abstract method based on improvement power derivation graph layout, step is as follows：

1. preliminary placement is generated using improved power guiding algorithm.Improved power guiding algorithm be exactly first with FR algorithms from Beginning is put in seat in the plane, and iterate searching state of minimum energy, produces a relatively stable layout, then perform based on the state LinLog algorithms.

1.1 initialization：Primary condition is set, each node v is assigned_iOne random site p_i

1.2 figures based on FR algorithms are laid out

To each node v in figure_iMake following iteration：

What 1.2.1 calculate node was subject to makes a concerted effort：A) first calculate node v_iWith other nodes v_jBetween Euclidean distance x=| p_i-p_j|.B) substitution formula (1) and (2) calculates the gravitation and repulsion from other nodes suffered by the node respectively.C) according to public affairs Formula (3) calculate node v_iSuffered F (the v that make a concerted effort_i)

f(e_ij)=x²/k (1)

g(v_i, v_j)=k²/x (2)

Wherein, k=sqrt (A/ | V |), A are the viewing area area of whole view, and | V | is node total number；e_ijRepresent section Point v_iWith v_jBetween the side that is connected, therefore formula (1) calculate two have the connected node in side between gravitation.Formula (2) calculates at 2 points Between repulsion.In formula (3)Represent the direction of power.

1.2.2 node location updates：Node v_iAccording to the effect F (v that make a concerted effort above_i) move, reach new position p_i'=p_i+C*F(v_i).Here C is common constant, is the influence coefficient of power.When C is larger, joints influence is big, motion Faster；Vice versa.

1.2.3 iteration termination condition is judged：The gross energy E of whole system, then calculate node iteration are calculated according to formula (4) Energy differences Δ E before and after location updating, when Δ E is continuously less than the threshold value that certain sets, then it is assumed that system reaches stable state, Stop iteration.

Figure layout under 1.3LinLog algorithms

Based on the layout of FR algorithms, the position of each node is recorded, to each node v in figure_iMake following iteration：

What 1.3.1 calculate node was subject to makes a concerted effort：A) first calculate node v_iWith other nodes v_jBetween Euclidean distance x=| p_i-p_j|.B) substitution formula (5) and (6) calculates the gravitation and repulsion from other nodes suffered by the node respectively.C) according to public affairs Formula (3) calculate node v_iSuffered F (the v that make a concerted effort_i)

f(e_ij)=C₁ (5)

g(v_i, v_j)=C₂/x (6)

C in formula₁And C₂All it is constant.

1.3.2 node location updates：Node v_iAccording to the effect F (v that make a concerted effort above_i) move, reach new position p_i′。

1.3.3 iteration termination condition is judged：The gross energy E of whole system, then calculate node iteration are calculated according to formula (7) Energy differences Δ E before and after location updating, when Δ E is continuously less than the threshold value that certain sets, then it is assumed that system reaches stable state, Stop iteration.

2. the layout result based on power guiding algorithm, generates the hierarchical structure of figure.

2.1 definition cluster c are five-tuple (l_c,r_c,w_c,f_c,V_c), wherein l_cAnd r_cTwo son clusters of cluster are represented respectively In left subgroup and right subgroup, w_cIts weights are represented, as (distance can select different degree to the distance between its sub- cluster Amount), f_cIt is his father's cluster, V_cThen represent original node set in the figure included in the cluster.

All ancestor node v in 2.2 initialization figures_iFor leaf clusters c_i(u,u,0,u,v_i), u is expressed as sky here, while The position Pc of initialization leaf cluster_iIt is the v after exertin guiding layout_iPosition p_i。

2.3 calculate the distance between cluster.According to the different abstract needs of network data and vision, can select it is different away from Calculated from measurement.

2.3.1 the Euclidean distance r between cluster is calculated_cicj=| Pc_i-Pc_j| as distance metric.

2.3.2 the average path length l between cluster is calculated_cicjAs distance metric.

2.3.3 according to formula (8) by the average path length l between cluster_cicjWith the Betweenness Centrality C of cluster_B(c_i) Sum is combined, used as distance metric.

Wherein α ∈ (0,1) are weighting parameter, represent the shared weight in weights of Betweenness Centrality.α is bigger, then intermediary Influence of the centrality to weights is bigger.Ω_ijIt is node v_iAnd v_jBetween shortest path bar number, Ω_ij(v_k) it is section in these paths Point v_kThe number of times of appearance, therefore, C_B(v_k) represent node v_kBetweenness Centrality.Cluster c_kBetweenness Centrality C_B(c_k) poly- for this The average value of the node Betweenness Centrality that class is included.

2.4 select two cluster c_iWith c_jApart from d_ijRecently, the new cluster c of generation is merged as left and right subgroup_k (c_i,c_j,d_ij,u,(V_ci∪V_cj)), position Pc_kThen according to the node set Vc that new cluster is included_kThe mean place of interior joint To determine.Meanwhile, newly cluster c_kAs c_iWith c_jFather cluster, c_iWith c_j(u, u, 0, c are updated to respectively_k,V_ci) and (u, u, 0, c_k,V_cj), and by c_kAdd and calculate.

2.5 repeat step 2.3 and 2.4 always, find closest cluster and merge repeatedly, until all clusters all Untill merging into a cluster.

3. it is abstract vision to be carried out in different levels.

3.1 will finally merge the cluster c for obtaining when building hierarchical structure_rootIt is defined as root cluster；

3.2 define abstract level parameter AL ∈ [0,1], with AL × w_crootHierarchical structure is intercepted to spend, only will symbol Conjunction condition w_ck≤AL×w_croot<w_fcCluster be shown on view, complete it is abstract to the vision of network structure.By abstract Level AL's changes to obtain different abstract results.

Example application below, the present invention will be described.In this example embodiment, apply the present invention to data in literature, represent Cooperative relationship and pattern between author, step are as follows：

1st, every author of article is regarded into a node, and assigns unique ID v_iIt is identified；Complete a same piece There is cooperative relationship between the author of article, regard this cooperative relationship as a line, and assign unique ID e_ijEnter rower Know, represent author v_iWith author v_jBetween have cooperation.Similarly, the author of same piece article is connected with side between any two.Will be all Data in literature is switched to network structure by article after being processed as above.

2nd, the preliminary placement of literature author's cooperative relationship is generated using improved power guiding algorithm.Improved power guiding algorithm It is exactly that iterate searching state of minimum energy first with FR algorithms since random site, produces a relatively stable layout, LinLog algorithms are performed based on the state again.Distance is nearer between the layout interior joint of generation, represents conjunction between corresponding author Make relation closer.

2.1 initialization：It is that every author assigns random initial position p in visual two dimensional surface region_i(x,y)

2.2 figures based on FR algorithms are laid out

To every author v in article_iMake following iteration：

What 2.2.1 calculating author was subject to makes a concerted effort：A) author v is first calculated_iWith other author v_jBetween Euclidean distance x=| p_i-p_j|.B) substitution formula (1) and (2) calculates the gravitation and repulsion from other authors suffered by the author respectively.C) according to public affairs Formula (3) calculates author v_iSuffered F (the v that make a concerted effort_i)

f(e_ij)=x²/k (1)

g(v_i, v_j)=k²/x (2)

Wherein, k=sqrt (A/ | V |), A are the viewing area area of whole view, and | V | is author's sum；e_ijRepresent and make Person v_iWith v_jBetween have a cooperative relationship, therefore formula (1) calculates two gravitation having between the author of cooperative relationship.Formula (2) is calculated Repulsion between two authors.In formula (3)Represent the direction of power.

2.2.2 author's location updating：Author v_iAccording to the effect F (v that make a concerted effort above_i) move, reach new position p_i'=p_i+C*F(v_i).Here C is common constant, is the influence coefficient of power.When C is larger, author's stressing influence is big, motion Faster；Vice versa.

2.2.3 iteration termination condition is judged：The gross energy E of whole system is calculated according to formula (4), then calculates author's iteration Energy differences Δ E before and after location updating, when Δ E is continuously less than the threshold value that certain sets, then it is assumed that system reaches stable state, Stop iteration.

Figure layout under 2.3LinLog algorithms

Based on the layout of FR algorithms, the position of each author is recorded, to each author v_iMake following iteration：

What 2.3.1 calculating author was subject to makes a concerted effort：A) author v is first calculated_iWith other author v_jBetween Euclidean distance x=| p_i-p_j|.B) substitution formula (5) and (6) calculates the gravitation and repulsion from other authors suffered by the author respectively.C) according to public affairs Formula (3) calculates author v_iSuffered F (the v that make a concerted effort_i)

f(e_ij)=C₁ (5)

g(v_i, v_j)=C₂/x (6)

C in formula₁And C₂All it is constant.

2.3.2 author's location updating：Author v_iAccording to the effect F (v that make a concerted effort above_i) move, reach new position p_i′。

2.3.3 iteration termination condition is judged：The gross energy E of whole system is calculated according to formula (7), then calculates author's iteration Energy differences Δ E before and after location updating, when Δ E is continuously less than the threshold value that certain sets, then it is assumed that system reaches stable state, Stop iteration.

3rd, the result based on power guiding algorithm layout, generates the hierarchical structure of literature author's cooperative relationship.

3.1 define consortium c for five-tuple (l_c,r_c,w_c,f_c,V_c), wherein l_cAnd r_cTwo sons of difference representative organization Left group and right group in group, w_cRepresent its weights, the as distance between its sub- group, f_cIt is his father group, V_cThen table Show the author set included in the group.

3.2 regard all authors as one by one group, initialize all author v_iIt is initial group c_i(u,u,0,u, v_i), an author is only included in each group in an initial condition.Here u is expressed as sky, while initializing the position of initial group Put Pc_iIt is the v after exertin guiding layout_iPosition p_i。

3.3 calculate the distance between group.In calculating average shortest path length two-by-two between author and every intermediary of author Disposition, according to formula (8) using the average path length between group and Betweenness Centrality sum as distance metric.

Wherein α ∈ (0,1) are weighting parameter, represent the shared weight in weights of Betweenness Centrality.α is bigger, then intermediary Influence of the centrality to weights is bigger.Ω_ijIt is author v_iAnd v_jBetween shortest path bar number, Ω_ij(v_k) it is work in these paths Person v_kThe number of times of appearance, therefore, C_B(v_k) represent author v_kBetweenness Centrality.Group c_kBetweenness Centrality C_B(c_k) it is the group The average value of author's Betweenness Centrality that body is included.

3.4 select Liang Ge groups c in all entities_iWith c_jApart from d_ijRecently, closed as left and right group And produce new consortium c_k(c_i,c_j,d_ij,u,(V_ci∪V_cj)), position Pc_kThe author's set Vc for then being included according to new attribute_k The mean place of middle author determines.Meanwhile, new consortium c_kAs c_iWith c_jFather group, c_iWith c_jBe updated to respectively (u, u,0,c_k,V_ci) and (u, u, 0, c_k,V_cj), and by new attribute c_kAdd and calculate.

3.5 repeat step 3.3 and 3.4 always, find closest group and merge repeatedly, until by all of work Untill person group merges into a group.

4th, to carry out vision in different levels abstract.Abstract level parameter is adjusted, shows that the author of different abstract levels closes Make relational structure.

4.1 will finally merge the consortium c for obtaining when building hierarchical structure_rootIt is defined as total group

4.2 define abstract level parameter AL ∈ [0,1], with AL × w_crootHierarchical structure is intercepted to spend, only will symbol Conjunction condition w_ck≤AL×w_croot<w_fcConsortium be shown on view, complete to the vision of literature author's cooperative relationship structure It is abstract.Different abstract results are obtained by the change of abstract level AL.When abstract level is 0, it is shown that power guiding cloth Office's result, as abstract level increases, cooperates close author and gradually merges into consortium, while during some have an intermediary high The author of disposition is retained.Represent the modality for co-operation of literature author by abstract view, such as individually creation, groupuscule are closed Work, on a large scale across group cooperation etc..

Claims (1)

1. a kind of based on the level vision abstract method for improving power derivation graph layout, step is as follows：

Step 1. generates preliminary placement using improved power guiding algorithm；Improved power guiding algorithm be exactly first with FR algorithms from Beginning is put in seat in the plane, and iterate searching state of minimum energy, produces a relatively stable layout, then perform based on the state LinLog algorithms；

1.1 initialization：Primary condition is set, each node v is assigned_iOne random site p_i

1.2 figures based on FR algorithms are laid out

To each node v in figure_iMake following iteration：

What 1.2.1 calculate node was subject to makes a concerted effort：A) first calculate node v_iWith other nodes v_jBetween Euclidean distance x=| p_i-p_j |；B) substitution formula (1) and (2) calculates the gravitation and repulsion from other nodes suffered by the node respectively；C) according to formula (3) calculate node v_iSuffered F (the v that make a concerted effort_i)

f(e_ij)=x²/k (1)

g(v_i,v_j)=k²/x (2)

$F\left(v_i\right)=\underset{e_{ij}}{\&Sigma;}f\left(e_{ij}\right)\stackrel{\&OverBar;}{x}-\underset{i\&NotEqual;j}{\&Sigma;}g(v_i,v_j)\stackrel{\&OverBar;}{x}---\left(3\right)$

Wherein, k=sqrt (A/ | V |), A are the viewing area area of whole view, and | V | is node total number；e_ijRepresent node v_i With v_jBetween the side that is connected, therefore formula (1) calculate two have the connected node in side between gravitation；Formula (2) calculates point-to-point transmission Repulsion；In formula (3)Represent the direction of power；

1.2.2 node location updates：Node v_iAccording to the effect F (v that make a concerted effort above_i) move, reach new position p_i′ =p_i+C*F(v_i)；Here C is common constant, is the influence coefficient of power；When C is larger, joints influence is big, motion compared with Quickly；Vice versa；

1.2.3 iteration termination condition is judged：The gross energy E of whole system, then calculate node iteration position are calculated according to formula (4) The energy differences Δ E before and after updating is put, when Δ E is continuously less than the threshold value that certain sets, then it is assumed that system reaches stable state, stops Only iteration；

$E={\mathrm{\&Sigma;}}_{e_{ij}}|p_i-p_j{|}^3-{\mathrm{\&Sigma;}}_{i\&NotEqual;j}\ln |p_i-p_j|---\left(4\right)$

Figure layout under 1.3 LinLog algorithms

Based on the layout of FR algorithms, the position of each node is recorded, to each node v in figure_iMake following iteration：

What 1.3.1 calculate node was subject to makes a concerted effort：A) first calculate node v_iWith other nodes v_jBetween Euclidean distance x=| p_i-p_j |；B) substitution formula (5) and (6) calculates the gravitation and repulsion from other nodes suffered by the node respectively；C) according to formula (3) calculate node v_iSuffered F (the v that make a concerted effort_i)

f(e_ij)=C₁ (5)

g(v_i,v_j)=C₂/x (6)

C in formula₁And C₂All it is constant；

1.3.2 node location updates：Node v_iAccording to the effect F (v that make a concerted effort above_i) move, reach new position p_i′；

1.3.3 iteration termination condition is judged：The gross energy E of whole system, then calculate node iteration position are calculated according to formula (7) The energy differences Δ E before and after updating is put, when Δ E is continuously less than the threshold value that certain sets, then it is assumed that system reaches stable state, stops Only iteration；

$E={\mathrm{\&Sigma;}}_{e_{ij}}|p_i-p_j|-{\mathrm{\&Sigma;}}_{i\&NotEqual;j}\ln |p_i-p_j|---\left(7\right)$

Step 2. is based on the layout result of power guiding algorithm, generates the hierarchical structure of figure；

2.1 definition cluster c are five-tuple (l_c,r_c,w_c,f_c,V_c), wherein l_cAnd r_cIn representing two of cluster son clusters respectively Left subgroup and right subgroup, w_cRepresent its weights, the distance (distance can select different measurements) as between its sub- cluster, f_c It is his father's cluster, V_cThen represent original node set in the figure included in the cluster；

All ancestor node v in 2.2 initialization figures_iFor leaf clusters c_i(u,u,0,u,v_i), u is expressed as sky here, while just The position Pc of beginningization leaf cluster_iIt is the v after exertin guiding layout_iPosition p_i；

2.3 calculate the distance between cluster；According to the different abstract needs of network data and vision, different distances can be selected Measurement is calculated；

2.3.1 the Euclidean distance r between cluster is calculated_cicj=| Pc_i-Pc_j| as distance metric；

2.3.2 the average path length l between cluster is calculated_cicjAs distance metric；

2.3.3 according to formula (8) by the average path length l between cluster_cicjWith the Betweenness Centrality C of cluster_B(c_i) it Be combined, as distance metric；

$w_{c_k}=\mathrm{\&alpha;}\&times;\left(C_B\right(c_i)+C_B(c_j\left)\right)+(1-\mathrm{\&alpha;})\&times;l_{c_i{c}_j}---\left(8\right)$

$C_B\left(c_k\right)=\frac{1}{\left|{\mathrm{Vc}}_k\right|}*{\mathrm{\&Sigma;}}_{v_k\&Element;{\mathrm{Vc}}_k}{C}_B\left(v_k\right)---\left(9\right)$

$C_B\left(v_k\right)=\underset{i\&NotEqual;j\&NotEqual;k}{\&Sigma;}\frac{{\mathrm{\&Omega;}}_{ij}\left(v_k\right)}{{\mathrm{\&Omega;}}_{ij}}---\left(10\right)$

Wherein α ∈ (0,1) are weighting parameter, represent the shared weight in weights of Betweenness Centrality；α is bigger, then intermediary center Influence of the property to weights is bigger；Ω_ijIt is node v_iAnd v_jBetween shortest path bar number, Ω_ij(v_k) it is these paths interior joints v_kGo out Existing number of times, therefore, C_B(v_k) represent node v_kBetweenness Centrality；Cluster c_kBetweenness Centrality C_B(c_k) wrapped by the cluster The average value of the node Betweenness Centrality for containing；

2.4 select two cluster c_iWith c_jApart from d_ijRecently, the new cluster c of generation is merged as left and right subgroup_k (c_i,c_j,d_ij,u,(V_ci∪V_cj)), position Pc_kThen according to the node set Vc that new cluster is included_kThe mean place of interior joint To determine；Meanwhile, newly cluster c_kAs c_iWith c_jFather cluster, c_iWith c_j(u, u, 0, c are updated to respectively_k,V_ci) and (u, u, 0, c_k,V_cj), and by c_kAdd and calculate；

2.5 repeat step 2.3 and 2.4 always, find closest cluster and merge repeatedly, until all clusters are all closed And untill being clustered for one；

It is abstract that step 3. carries out vision in different levels；

3.1 will finally merge the cluster c for obtaining when building hierarchical structure_rootIt is defined as root cluster

3.2 define abstract level parameter AL ∈ [0,1], with AL × w_crootHierarchical structure is intercepted to spend, only will meet bar Part w_ck≤AL×w_croot<w_fcCluster be shown on view, complete it is abstract to the vision of network structure；By abstract level AL's changes to obtain different abstract results.