Unified Mapping of Social Networks into 3D Space

Jiaqian Zheng and Junyu Niu Dept. of Computer Science, Fudan University 200433 Handan Road, Shanghai, P.R.C

baterhome@gmail.com; jyniu@fudan.edu.cn

Abstract Social network visualization focuses on nodes

lacking inherent layout semantics. Many previous mapping algorithms locating graph data into physical space are based on assumptions that space is in two -dimensions and that the visualization of huge graphs can be realized by dividing them into subgraphs independently for easier exploration.

However, when physical space extends to three dimensions, the automatic layout becomes a problem. Different from sparse graphs, nodes in social networks may be closely connected to each other, which raises another problem that how to explore them in undivided ways. For these reasons, the primary motivation for this work is to provide a unified framework optimized for social network a 3D exploration by combining techniques widely used in data mining and computational graphics.

In the framework, we borrow octree structure to solve nodes self-adapting layout problem in 3D space and propose an algorithm to map a social network into an octree without losing connection information after space division, and define two practical operations (shifting and shading) to satisfy the details-on-demand requirement of data visualization.

By building a prototype system on Java 3D, the experiment visualizes the social networks extracted from real organizations and demonstrates the flexibility of the framework. Our methods produce a satisfying performance. Keywords: visualization, social network, layout, user interface 1. Introduction

Computer systems today store vast amounts of data. Researchers at the University of California, Berkeley, estimated that about 1 exabyte (1 million terabytes) of data is generated annually worldwide, including 99.997% only in digital form. This worldwide data deluge means that, in the next three years, more data will be generated than during all previous human history [10].

The visual data exploration process can be viewed as a hypothesis-generation process, whereby, through

visualizations of the data. It allows users to gain insight into the data and come up with new hypotheses. Verification of the hypotheses can also be accomplished via visual data exploration as well as through automatic techniques derived from statistics and machine learning.

Graphics appear in numerous applications such as web browsing, state-transition diagrams, and data structures. The ability to visualize and to navigate in these potentially large, abstract graphs (such as social networks that contain hundreds nodes) is often a crucial part of an application.

Information visualization has specific challenges: 1) the capability to explore large datasets within the accepted system performance, which means there should be a macroscopic overview representing the whole datasets; 2) how to drills down to access details, the methods could be interactive ones as well as definitions to query subgraphs[14]; 3) reconstructing layout information of the data. Lots of related researches followed these challenges but most of them implement the process of visualization only in 2D physical space.

However, with the development of computer hardware, computational ability rises to sustain more expressive human interfaces such as multimedia and 3D render techniques. Then it raises a new question: how to solve challenges listed above within the environment of 3D space. Formerly, The unified 3D visualization problem (abbreviated to U3V Problem below) is defined as follows.

Input: An undirected social network G(V,E) Listed weights of each edges W(E)

Range of X,Y,Z dimensions in 3D space

Output: Position(x,y,z) of each vi in V Constraint: Represent closeness intuitively Accepted performance Details on demand

Closeness can be regarded as how relatively nodes

connect to each other. It will be formally defined in a later section. To address the problem, the main contributions of this work include: 1) Employ an octree structure to cut space and arrange positions for each node; 2) Propose a quick mapping algorithm of social

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networks into octree structure; 3) Integrate the exploration of whole social network between the overview and details into one view by two operations - shifting and shading.

This paper is organized as follows: in Section 2 we briefly introduce related work in social network visualization. In Sections 3, 4 and 5, we discuss our approaches to three components in the framework. Section 6 gives the experiment and the visual results of prototype system. We analyze the results in section 7 and give our conclusion. 2. Related Work

Visual data exploration has several main advantages [10] over the automatic data mining techniques in statistics and machine learning. It helps analysts deal more easily with highly inhomogeneous and noisy data since it is intuitive to human understanding. Another reason is that it requires no understanding of complex mathematical or statistical algorithms or parameters.

So there are many sophisticated applications based on social network visualization such as criminal network detection[1][2] and police intelligence[5]. Organized crimes are carried out by multiple collaborating offenders who may form groups and teams and play different roles. A clear understanding of these structural properties in the visual form of criminal networking may help analysts target critical network members for removal or surveillance, and locate network vulnerabilities where disruptive actions can be effective.

One of basic work to the visual social networks is to determinate the position of each node. One approach employs a hyperbolic tree metaphor to visualize crime relationships [8]. It is especially helpful for visualizing a large amount of relationship data because it simultaneously handles both focus and context. Another approach uses a spring embedder algorithm[9] to adjust the positions of nodes automatically to prevent a network display from being too cluttered in 2D space.

Based on researches in 2D environments, some related work[4] expand the process of visualization to 3D environments. However visualization in 3D space may easily reach bottleneck of system performance when the amount of graph data increases rapidly. Cone trees[3] is the satisfied solution to handle huge amount by representing data in hierarchical structure. In the process only visible parts of a hierarchical data can be involved into computation, which reduces the time cost in the rendering stage. It shares same ideas with [12]. However there is not a provided efficient mapping algorithm of social networks into cone trees yet.

Regardless of dimensions, there are two crucial steps to realize the visualization. First one is to map social networks into hierarchical structures explicitly, which makes final layout results produced from visualization properly correspond to social connections. Since social network graphs are usually not quite sparse and nodes in them could be widely connected to each other, that makes general graph partition algorithms like NCut[12], or Min-Max Cut[13] meaningless. One reason is that the process to extracting hierarchical structures in visualization does not pursuit high precision. Improved solutions can employ a relatively simplified algorithm to leverage higher efficiency during the 3D rendering. Another reason is that mapping hierarchical structures by iteratively cutting the graphs is not cost-efficient. Referring to the complexity analysis in [16], we need a new, lighter mapping algorithm. In the second step, by inputting the result of mapping, a layout algorithm may help to generate layout information for each entity in the social network by visiting the hierarchical structure from root to leaves.

In the next three sections, we firstly discuss the solution to automatic 3D layout problem and find out the data structure as its input; then explore algorithm mapping the social network into this structure; and, finally, practice visualization with operations provided by framework. 3. Approach to 3D Layout 3.1 Space division

The first step to addressing layout problem is to decide space division function. There are some constraints for the selection of space division methods to correspond with the requirements of unified visualization:

1. The physical distance between two entities could

intuitively represent how closely they are related to each other in the social network.

2. Division methods would inherently provide enough supports leading to easy implementation of navigating the details-on-demand.

3. Not quite complex enough to ensure efficiency in 3D rendering.

So we borrow the octree algorithm[6] for its specific

features. The rendering of objects in 3D physical space using octrees has been used frequently in the field of computer graph. Octrees involve the subdivision of the space into a set of small cells. Each cell is called octant and represents these octants in the form of a tree recursively. The maximum out degree of each octant

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comes to eight, which strictly corresponds to 3D space division as Figure 1a illustrated.

Figure 1. Layout step in framework

Once an octree (see in Figure 1b) has been generated

(to be discussed in next section), a layout algorithm can be used. Each leaf in octree denotes a single entity in social network, and non-leaf node denotes 3D space boundary which preserves room for its children. 3.2 Layout algorithm

Associated with the space division function at a layout algorithm can be given in algorithm 3.1. For each of these octants there is a code to determine boundary of objects and each of node is represented in same format and constructed together recursively, an octree well satisfies the Constraint 2 listed above. Moreover, many related researches[15][11] demonstrate the good complexity of its algorithm in rendering, so it can meet Constraint 3 as well.

ALGORITHM 3.1 Layout Input: nodep, root of octree without position

Boxp , 3D space boundary to be allocated Output: nodep, position field of each son in it is filled nodep.pos = center-point of Boxp if nodep is not leaf for i ← [1..8] if nodep.son[i] is not null Boxi = sub-box of Boxp in ith octent direction AlgorithmLayout (nodep.son[i], Boxi) Return nodep

Before evaluating Constraint 1, we have to explain

how to describe the measurement of closeness. The U3V problem, defined in last section, network graph G(V,E) is available at beginning. After hierarchical mapping, V is formatted into an octree(figure 1b, the mapping methods are discussed in the next section). Then the closeness of any two nodes in the octree can be defined as follows:

DEFINITION 3.1 iLN denotes node N in the ith octant

of its direct parent and in level L away from octree root.

DEFINITION 3.2 Represent G(V,E) in format of symmetric matrix |||| VVM × , If both i

LN 1 and jLN 2 are

leaves, the closeness between them can be defined as ),cos(),( 21 ji

jL

iL MMNNC =

if anyone, supposing iLN 1 is non-leaf node

∑=

+=n

k

jL

kL

jL

iL NNC

nNNC

121121 ),(1),(

where iLN 1 is the direct parent of all k

LN 11+ , n is the

count of iLN 1 ’s sons.

In the above definition, Mi and Mj denote vectors at

line i and line j of M respectively. If neither iLN 1 nor

jLN 2 is leaf, the closeness can also be calculated by

decomposing one of them into leaves in turn. The definition exposes the underlying rule that the more similar two nodes connected with others, the more closely they related to each other. Note that the edges directly connect two adjoin nodes together also take effect in closeness calculation.

Figure 1b is generated by a mapping algorithm; the bigger ),( 21

jL

iL NNC is, the earlier i

LN 1 and jLN 2

combined into one octant node. Then Algorithm 3.1 can comply with Constraint 1. The next section will focus on this mapping algorithm. 4. Light Mapping

This section tackles on how to map a social network quickly into an octree structure described in previous section.

Figure 2: Mapping step in framework

As the step illustrated in Figure 2b, and 2c, in order

to make layout step produce optimal 3D visualization, mapping methods should be restricted by Constraints 1. However the algorithm used in this work only reaches local optimization for the following two reasons: 1) Visualization does not require a strict precision of hierarchical mapping; and 2) It's hardly to find the global optimization solution. Algorithm 4.1 provides a

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quick approach to convert social networks into octrees simply by employing a greedy search.

ALGORITHM 4.1 Mapping Input: Set SN{Vi}, all nodes in a social network

Weighted Closeness set SC{<Vi,Vj,closeness>}

Output: Octree TOT While |SN|>1 <Vm, Vn, closeness> = MAXcloseness (SC) degree = count (Vm.son) + count (Vn.son) set Vk to be anew empty node if degree <=8 move Vm.son, Vn.son to Vk.son else add Vm, Vn to Vk.son remove Vm, Vn from SN, add Vk into SN remove all edges related to Vm, Vn from SC by definition 3.2, get all non-zero weighted- closeness {<Vk,Vi, closeness>} and put them into SC Return the only element of SN

In Algorithm 4.1, SN{Vi} contains all leaves nodes

initially where |SN{Vi}|= N. With the running of algorithm, two nodes will be removed and one non-leaf node added during each iteration. There is no need to ensure the out degree of each node to be eight.

Complexity of algorithm: The main cost in loop is the step to update weighted edges SC. The definition 3.2 is recursive and cost for C(VK,Vi) comes up to depth of VK and Vi, so the selection of all non-zero weighted edges {<Vk,Vi,closeness>} comes to N × logN. However, the useful calculation of the closeness of each non-leaf node is only once. Caching the closeness result can reduce the complexity to (N-1)+logN×1 per loop. Finally, we conclude the complexity of Algorithm 4.1 at O(N×N). 5. Navigating

Figure 3. Depth vs. Detail

After combining the mapping and layout steps, we

have mainly solved the U3V problem. The remaining

part of social network visualization is to meet details-on-demand, which is associated with two problems: how deeply the visualization process drills down and how many details a social network exposes to the visible horizon. The difference between depth and detail can be seen in Figure 3.

As Figure 3 illustrates, the input value let the

framework make the decision from which octant node begins to render. The bigger this value is, the closer hierarchy is displayed. Corresponding to the concept of the depth, the value of detail helps user appoint the level of detail (LOD) to be exposed. Furthermore, we give the definitions of two operations acting on them:

DEFINITION 5.1 Shifting denotes operations increasing or decreasing value of depth. DEFINITION 5.2 Shading denotes operations enlarging or shortening level of details.

The framework provides overview and filter methods to users by shifting operations by adjusting the depth value interactively, since connection related to octant nodes above the depth will be filtered away. In contrast, shading is a kind of hierarchical operation without filtering away any connections after space division. Even the octant nodes out of the LOD range, the social network relations can be displayed by forming a merged, weighted linking to one of their visible parents. In display, merged weights are calculated by the recursive definition 3.2. In other words, shading on value of LOD enables the framework to support zoom in and zoom out in data visualization.

ALGORITHM 5.1 LODD, level of depth and details Input: nodep, one node of octree

SOC = <o1,o2,…,on>, sequence of octants dV, details of visualization

Output: CVnodes, collection of visible nodes if SOC is not empty oi = first octant of SOC SOC = SOC - oi CVnodes = Algorithm LODD (nodep.son[oi], SOC, dV) else if dV = 0 or nodep is leaf CVnodes = { nodep } else for each nodei in nodep.son CVnodes = CVnodes ∪ Algorithm LODD (node i , {}, dV -1) Return CVnodes

Here is the Algorithm 5.1 implements shifting and

shading operations, where depth is given in the form of

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the sequence, including all past octant nodes from root to target node, and the LOD is appointed by a scalar integer.

In LODD, algorithm will return a collection containing either leaf nodes or non-leaf nodes. Non-leaf nodes represent the hierarchical structures of leaf nodes out of LOD. In the framework, they are displayed by different 3D-geometries from others. 6. Experiment and Result

We use the dataset from TREC 2006 Expert Search task[18] (abbreviated as ES06 below) to evaluate our framework. ES06 provides us an expert candidate list including each candidate's full name and official E-mail address, and another E-mail corpus crawled from practical W3C networks.

The ES06 contains more than 300 experts with over 10000 E-mail transactions. We construct the social network from E-mail records as following steps: a) Extract expert names, set Vexpert from <From> and <To> fields; b) For each mail, add an undirect link with weight 1 between experts appeared in <From> and <To> fields. If there is already a link, just increase the weight by 1. c) Return an indirect graph G(V, E) representing the whole social network.

We implement a prototype system of the framework on Java 3D[17] to explore real social networks. The explorer takes the E-mail corpus of ES06 as input, and outputs a uniform 3D graph on the screen. Users can insight into the graph from any distance and view points by moving, wheeling or dragging the mouse in 3D space.

Table 1: Geometry meanings

Geometry Color Definitions sphere yellow visible experts in ES06 cone white email relation between experts axes rgb coordinates axes in 3D space box 3D yellow represent experts out of LOD L shape purple △ shape cyan same region in different depth

□ shape orange □ shape cyan same region in different LOD

Table 1 lists the meanings of each geometry in the

experiment results. All 3D objects are rendered by the explorer automatically but 2D geometries with a doted line drawn on manually to intensify effects of shifting and shading operations. Cone represents E-mail relations, and the higher weight value one relation contains, the thicker the cone displays.

First, we demonstrate the shifting operations in appendix figure 4. Notice that Figure 4b is the expansion of Figure 4a in 6th octant direction. The connections related to other octants have been removed by the filter operation. Similarly, figure 4c is the expansion at 1st octant. Users can easily distinguish single experts after down-shifting into the network.

Then, figure 5(d,e,f) illustrates the results of shading operations at LOD of (2,3,4) respectively. With the increment of LOD value, more expert entities in the network can be visible out of the hierarchical boxes. The combined connection to one hierarchical box will decomposes to detailed ones which are reconnected to those new expert nodes.

Overhead of our framework: Based on the implementation of techniques and the algorithms discussed in previous sections, the framework runs our measurements on an ThinkPad T43 PC with one 1.86 GHz processor, 512 MB main memory, one 80 GB disk, and running the Microsoft Windows XP operating system. The shifting operation always costs less than one second in any depth, and the rotation, movement and zoom in the 3D environment run, smoothly at LOD less than 5. 7. Conclusions

In this work, we define and address the problem of the U3V Problem. The major contribution of this paper is to give a unified framework automatically visualizing social networks into 3D space.

In the process, our framework employs octrees to solve the automatic 3D layout problem and yields satisfactory rendering performance. To comply with requirements of overview, details, zoom and details-on-demand in data visualization, we carry out two novel practical operations shifting and shading to navigate social networks intuitively without breaking them into isolated entities. This keeps the integrality of relations between each entities in the network. Also, to flexibility, our method discards the underlying assumption that graph data can easily convert to tree format, takes mapping as a part of the framework into consideration, and the leverage mapping algorithm between effect and efficiency which optimizes the framework for social networks’ visualization.

Moreover, we implement our methods in a working prototype, complete with a 3D space exploration interface, on a real social network derived from W3C working groups. The graph contains hundreds of leaf nodes.

Directions for future research include a mapping algorithm that contains more optimized features performed for intuitive social networks exploration; and

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random space division functions providing more flexible solutions to the automatic 3D layout problem.

Acknowledgements

We would like to thank Cherry Lin and Yao for their works in providing extracted social networks for experiment and Juan Xu for his advices on mapping in our discussion. 8. References [1] Jennifer Xu and Hsinchun Chen, “Criminal network analysis and visualization,” Communication of the ACM, June 2005, Vol. 48, No. 6. [2] M.K. Sparrow, “The application of network analysis to criminal intelligence: An assessment of the prospects,” Social Network, 13 (1991), pp. 251–274. [3] G.G. Robertson, J.D. Mackinlay and S.K. Card, “Cone Trees: animated 3-D visualizations of hierarchical information,” ACM SIGCHI Conference on Human Factors in Computing Systems, April 1991. [4] I. F. Cruz and J. P. Twarog, “3D graph drawing with simulated annealing,” Graph Drawing'95, LNCS vol.1027, pp. 162-165, Springer Verlag, 1996. [5] V. E. Krebs, “Mapping networks of terrorist cells,” Connections, 24, 3, pp. 43–52, 2001. [6] D. Ayala, P. Brunet, R. Juan and I. Navazo, “Object representation by means of nonminimal division quadtrees and octrees,” ACM Transactions on Graphics (TOG) Jan 1985, Vol. 4, No.1. [7] K. C. Cox, S. G. Eick and T. He, “3D geographic network displays,” Sigmod Record, Vol. 24, No 4, December 1996. [8] H. Chen, D. Zeng, Atabakhsh, W. Wyzga and J. Schroeder, “COPLINK: Managing law enforcement data and knowledge,” Commun. ACM, 46, 1, pp. 28-34, Jan 2003. [9] P. Eades, “A heuristic for graph drawing,” Congressus Numerantium, 42 (1984), pp. 149- 160. [10] D. A. Keim, “Visual exploration of large data sets,” Communications of the ACM, August 2001, Vol. 44, No. 8 [11] J. Veenstra and N Ahuja, “Line drawings of octree-represented objects,” ACM Transactions on Graphics (TOG) Jan 1988, Vol. 7, No. 1 [12] J. Shi, J. Malik, “Normalized cuts and image segmentation,” IEEE Transactions of Pattern Analysis and Machine Intelligence, Vol. 22, No. 8, August 2000. [13] C. Ding, X. He, H. Zha, M. Gu, H. Simon, “Spectral min-max cut for graph partition and data clustering,” ICDM 2001 [14] C. Faloutsos, K. S McCurley and A. Tomkins, “Fast discovery of connection subgraphs,” KDD, August 22-25, 2004 [15] J. Wilhelms and A. Van Gelder, “Octrees for faster isosurface generation,” ACM Transactions on Graphics(TOG) Jul 1992, Vol. 11, No. 3 [16] T. Feder, P. Hell, S. Klein and R. Motwani, “Complexity of graph partition problems,” STOC, Atlanta GA USA, 1999 [17] https://java3d.dev.java.net [18] I. Soboroff, A.P de Vries and N. Craswell, “Overview of the TREC 2006 Enterprise Track, TREC 2006, pp. 28

Appendix

In order to save spaces on paper, we merge 6 figures of experiment results into two groups. Each of them is represented row by row tightly, so the sequent letters of numbering are omitted below each figure. Instead, all are written together at button of page.

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Figure 4: Shifting operations (a,b,c)

Figure 5: Shading operations (d,e,f)

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