Border detection in complex networks

Bruno A N Travençolo¹, Matheus Palhares Viana¹ and Luciano da Fontoura Costa^1,2

¹ Institute of Physics of São Carlos—University of São Paulo, Avenue Trabalhador São Carlense 400, Caixa Postal 369, CEP 13560-970, São Carlos, São Paulo, Brazil
² National Institute of Science and Technology for Complex Systems, Brazil

E-mail: luciano@if.sc.usp.br

Received 6 March 2009
Published 12 June 2009

Contents

• 1. Introduction
• 2. Materials and methods
  • 2.1. Basic concepts
    • 2.1.1. Complex networks
    • 2.1.2. Diversity entropy
  • 2.2. Border detection
    • 2.2.1. A theoretical approach
  • 2.3. Border detection using other measurements
  • 2.4. Datasets
• 3. Results
  • 3.1. Detected borders using diversity entropy
  • 3.2. Extension for non-geographical networks
  • 3.3. Border detection for a denser network
  • 3.4. Border detection in a rewired network
  • 3.5. Detected borders using other measurements
  • 3.6. Real-world examples
• 4. Conclusions
• Acknowledgments
• References

1. Introduction

Complex networks have progressed all the way from the initial topological characterization of the Internet and WWW scale free properties (e.g. [1, 2]) to becoming a well-established and formalized research area (e.g. [3]–[5]) with myriad applications (e.g. [6]). Yet, given the relatively recent history of this field, there are still several fundamental aspects that deserve further attention from the complex networks community. The current work addresses one such fundamental and largely overlooked aspect, namely the problem of defining and identifying the borders of several types of networks. We should make clear at the outset that there is no formal definition of the borders of networks, so that their identification is intrinsically related to the individual definition of that concept.

Recent works [7, 8] have proposed the use of the entropy of the transition probabilities between nodes, with respect to dynamics such as random walks and self-avoiding random walks, in order to quantify the diversity of access of individual nodes. The idea underlying such an approach, like several other entropy-based methodologies, is conceptually simple and powerful. It is illustrated in figure 1 with respect to a simple graph. Though in (a) node 1 can access four nodes after two steps along a self-avoiding random walk, the transition probabilities to each of these nodes is markedly different. In the situation illustrated in figure 1(b), node 1 can also access four nodes, but with equal transition probabilities. The effectiveness of access from node 1 to the other nodes can be nicely and effectively quantified in terms of the entropy of the transition probabilities to the accessible nodes [8], implying the situation in (b) to be much more balanced and effective than the situation in (a). Actually, it can be shown that the minimal time for accessing all the nodes reachable from a given reference node after h steps is minimal when the entropy is maximum. Such a basic principle of the diversity entropy concept is adopted in the present work in order to define the borders of networks, in the sense that the diversity would be directly related to the internality of the nodes.

Though the relationship between diversity entropy and the internality of the nodes has been hinted at previously [7, 8], the present work reports a systematic investigation of such a definition of the borders of networks from several perspectives while taking into account model (theoretical) and real-world networks. After formally defining the concept of internality of each individual node in any network, we provide some analytical motivation with respect to diffusion (i.e. random walks) in regular lattices. Then we focus on geographical networks, whose borders can be intuitively related to their geographical structure (everybody has a conceptual idea of the borders of a city, for example). It is shown that the geographical borders tend to be in close agreement with the topological borders identified by the diversity entropy for both two-dimensional (2D) and 3D geographical structures. Then, we investigate how the concept of internality extends to non-geographical networks with respect to knitted networks [9], which are regular but non-geographical stochastic structures. We perform this study by making `holes' in the original networks, so that the borders of the holes can be well-defined topologically as corresponding to the limits of the holes. We show that the diversity entropy approach can precisely identify the original borders created by the holes. Subsequently, we take into account rewired (by using the procedure described by Maslov and Sneppen in [10]) versions of geographical networks, which allows us a series of insights into how the borders of initially geographically constrained networks change with perturbations. The important issue of quantifying the degree of possible correlations between the diversity entropy and well-established measurements such as the node degree and betweenness centrality are also investigated. These results corroborate little interrelationship between diversity entropy and such measurements, implying that the former feature is indeed providing additional information about the structure of the network. We conclude the current work by analyzing real-world networks, namely the plan of the town of São Carlos (SP, Brazil) as well as the network of word associations in Lewis Carroll's Alice's Adventures in Wonderland. Remarkable results are obtained for both cases, including the identification of modules of internality in the town and internality of words in Alice's Adventures in Wonderland.

2. Materials and methods

2.1. Basic concepts

2.1.1. Complex networks An unweighted and undirected complex network is a set of N nodes linked by E edges [3, 5] that can be described by the adjacency matrix A_{i, j}. This matrix has binary elements a_{i, j} that represent the presence or absence of connection between each pair of nodes. When a_{i, j} = 1 there is an edge between the node i and the node j. Otherwise, a_{i, j} = 0 indicates that there is no connection between them. The degree of the node i, k_i, is the number of nodes directly connected to i. This quantity is related to the adjacency matrix by the equation .

A walk of length h over the network is defined by a subset of h + 1 adjacent nodes (i.e. nodes that share at least one edge). Considering a discrete time Markovian process over the network, the transition probabilityP_h(j, i) is the probability that an agent departing from node i reaches the node j after h steps. This probability can vary according to the type of walk that is adopted. In the case of self-avoiding random walks, the moving agent cannot repeat nodes or edges during the walk [7, 8], whereas in traditional random walks the agent has no restrictions on performing the walk [11]. We used self-avoiding random walks for all the examples presented in this paper, as with this dynamics it is possible to reach more nodes in fewer steps. Two types of algorithms were used to compute the walks: a deterministic algorithm [12], in which all the possible walks on the network are evaluated and the respective probabilities are computed; and a probabilistic algorithm [8], where M self-avoiding random walks are performed in order to obtain the probabilities. The first algorithm is computationally more expensive, but is optimal and works well with networks with low average degree and low values of h. The second algorithm provides approximated values and is normally used for higher values of h and/or too large and densely connected networks.

2.1.2. Diversity entropy The diversity entropy of a particular node i measures how diverse is the access from this node to the other nodes of the network through a walk of length h. Let Ω be the set of all nodes of the network, except i, and N be the number of nodes of the network. The normalized diversity entropy E_h(Ω,i) can be expressed as [7, 8]:

The diversity entropy values for two simple graphs are shown in figure 1. We can note that the graph with balanced transition probabilities, i.e. figure 1(b), has higher diversity entropy.

2.2. Border detection

The concept of diversity is intuitively related to the definition of the border of a network. The main idea behind this property is that the peripheral nodes have not many options for proceeding on a random walk other than accessing the internal nodes of the network, resulting in low diversity values [8]. On the other hand, non-border nodes tend to have more effective and balanced access to most of the internal and peripheral nodes of the network, resulting in high diversity entropy values. In order to demonstrate this property analytically, we resorted to a bidimensional grid, as explained below.

2.2.1. A theoretical approach Let us consider a bidimensional semi-infinite grid illustrated in figure 2. We want to investigate the diversity entropy of the central node (shown in black in figure 2) located at distance d from the border of the grid. We start by evaluating the number of random walks of length h between the central node and all other reachable nodes. This number can be obtained by using a matrix W_h of size (h + 1)×(h + 1). Each entry (x, y) of this matrix represents the number of random walks of length h that reaches the element (p, q) of the grid (considering that the coordinate system is centered at the central node). The relation between (p, q) and (x, y) is given as p = 2x–h, q = 2y–h, where x≡y(mod h) and 0≤x,y≤h [13]. So, the number of random walks in the entry (x, y) is

where denotes the binomial symbol and

Note that the first term of this equation quantifies the total number of random walks in an infinite grid, while the second term (β_h(x, y)) quantifies the number of walks that are lost due to the barrier on the grid. The probability that a walker reaches the node at coordinate (p, q) after h steps starting at the central node is

Then, the diversity entropy of the central node considering random walks is

Observe that the sums run over all entries for which the value of P_h(x, y) is not null. Figure 2(b) shows the diversity entropy as a function of the distance d of the central node from the border, considered a fixed walk length step of h = 7. Note that in this figure the value of diversity entropy is lower near the border and constant when d > h (i.e. the central node does not suffer border effects).

2.3. Border detection using other measurements

In addition to the diversity entropy, we also evaluated the use of other measurements in order to define the border of networks. The considered features include: the node degree, the average shortest path length (ASPL), betweenness centrality (B) and the second eigenvector values [14]. The ASPL for the node i is the average length of all shortest paths between i and all other nodes of the network [5]. The betweenness centrality of the node i is defined as , where σ_{p, q} is the number of shortest paths from p to q and σ_{p, q}(i) is the number of shortest paths from p to q that go through i [5]. The second eigenvector method [14] takes into account the spectral properties of a dynamical process to find the extremities of the network.

2.4. Datasets

The efficiency of the proposed methodology was evaluated using three hypothetical geographical networks with different shapes and physical constraints, such as bottlenecks and internal holes. Geographical networks were used as they are the only type of network where the border nodes can be defined visually, allowing immediate evaluation of the accuracy of the border-detection methodology.

Figure 3 shows the geographical networks adopted in our analysis. The first network, depicted in (a), consists of a single square-shaped network. The case (b) is a network that includes bottlenecks (narrow connections) as well as internal holes. The third case, shown in (c), is a sphere-shaped 3D geographical network. In all cases, the nodes were generated by sampling spatial points in a space constrained by the bounding desired physical shape (e.g. points chosen within a circle of a given radius). The edges of the network were subsequently defined by Delaunay triangulation of the sampled points.

To validate the methodology for non-geographical networks, we performed the border detection over the knitted model [9]. This network is built from a set of non-connected nodes labeled from 1 to N. The basic step while growing this type of network involves shuffling the labels in an arbitrary sequence and defining the edges by connecting consecutive labels in that sequence. Note that the last label in the sequence is connected to the first one. This step is repeated n times. As result of this process, we obtain a non-geographical network where all nodes have degree k = 2n.

We also considered two real-world examples of networks: the first refers to the network of urban streets of the Brazilian town of São Carlos. In this network, each node represents a street crossing or beginning of routes, while the edges represent the streets (see [8] for additional information); the second is a word association network, built considering the relationship between words in the Lewis Carroll book Alice's Adventures in Wonderland. The nodes of this network represent the words, while the edges link words that are adjacent in the text [15].

3. Results

This section presents six different results of the border-detection methodology considering different measurements and data. First, the borders of the geographical networks presented in figure 3 were detected in terms of the diversity entropy and self-avoiding random walk dynamics. Next, the results of the proposed methodology applied to non-geographical networks are presented. In another analysis, we considered the border detection of a network four times denser than one of the networks of the first analysis. The next case shows an example of a geographical network that was progressively rewired and, again, the diversity entropy was used for border detection. Subsequently, the results of border detection using other measurements (node degree, betweenness centrality, average shortest path length and the second eigenvector) are shown and discussed. Finally, the results of border detection with respect to the two real-world networks are presented and interpreted.

3.1. Detected borders using diversity entropy

The diversity entropy values for the networks previously introduced were calculated considering five steps (h = 1,  ..., 5) and using the deterministic algorithm. The nodes of the networks shown in figures 3(a)–(c) were colored accordingly to their respective diversity entropy value for h = 3. The classification of a node as border or non-border can be performed by choosing a threshold for the diversity at a specific step. The nodes whose diversity entropy is below the threshold value are classified as border nodes. Figures 3(d)–(f) show the detected border nodes (green nodes). Notably, in all analyzed networks the border nodes were successfully identified.

In addition, it is important to note that the thresholds used to obtain these results were automatically chosen. The following procedure was adopted: first, the minimum distance from each node to the contour of the shape (the physical border) was determined. The nodes nearest the border (i.e. having low-distance values) were classified as border, whereas the others were understood as non-border nodes. Next, this classification was compared with the classification obtained by performing many consecutive thresholds of the diversity entropy value. The threshold value that results in the best relation between correctly classified border/non-border nodes was then used to define the border. Finally, note that this automatic procedure is only applicable to geographical networks. For more general networks, the user should supply a threshold based on some criterion such as a predetermined number of nodes, or a percentage of the total number of nodes.

3.2. Extension for non-geographical networks

In this section, we give some insights into the extensibility of the diversity to detect borders in non-geographical networks. Unlike the case for geographical networks, the definition of borders of non-geographical networks is not intuitive nor can it be visualized. In order to address such networks, we develop a methodology to assess the accuracy of the border-detection methodology, which basically consists of creating holes in a network. The main idea is to select a few nodes of the network and remove these nodes and their neighbors up to some predefined distance. By doing so, some holes in the network are created, and, as consequence, the neighbors of the removed nodes can be considered as border nodes. Then, after the identification of these border nodes, it is possible to evaluate the proposed border detection methodology by thresholding the diversity entropy and comparing the detected border with the actual borders defined by the holes.

This process is illustrated in figure 4. A knitted network with N = 300 and E = 600 is shown in figure 4(a). The hole in this network was obtained by removing the node with the highest diversity entropy value (note that other criteria could also be used) and its neighbors up to distance two. The border defined by this process is shown in figure 4(b) (green nodes), while the borders defined by thresholding the diversity entropy are shown in figure 4(c). Note that the results are quite good, with only a few misdetected nodes (4 nodes). For this analysis, the threshold was automatically chosen in a procedure very similar to that used in the geographical networks, with the difference that instead of using the distance from the physical border to define the actual border nodes, the nodes next to the hole were defined as the actual border.

In addition to the analysis of the border detection on a knitted network, we also tried the same procedure on a Barabási–Albert (BA) network [3]. However, the results obtained for a series of experiments showed that the ratio between correctly detected border/non-border were not satisfactory. The main reason for this result is that the average shortest path length in these networks is usually very low (e.g. this value is about 10 in a network with N = 1 000 000 and k = 4 [16]). As a consequence, the vast majority of the nodes in the network will be very close to the border defined by the holes, and their diversity entropy will be reduced (due to their proximity to the border). This reduction increases substantially the chances of misclassification. It is important to stress that the hole approach being unsuitable to evaluate the border detection methodology on BA networks does not invalidate our methodology. Given the other results reported in this work, as well as the motivation for defining the borders as nodes of low diversity, it is reasonable to expect that the method will also correctly identify the borders in small-world networks such as BA networks.

3.3. Border detection for a denser network

This section reports a comparison of the results of the proposed border-detection methodology considering networks with the same shape but with different density of nodes. For this purpose, we resorted to networks that contain narrow regions and holes (figure 3(b)), as well as versions of the same networks four times denser (with reference to nodes per square unit), shown in figure 5(a). For the latter case, the border-detection methodology was applied, i.e. the diversity entropy considering self-avoiding walks was calculated and a threshold on the diversity value was used to define the border (green nodes in figure 5).

The main difference between the results obtained for the two considered networks refer to the narrow regions of the shapes. The zoomed regions in figure 5 show some of these regions for both networks. While most of the nodes situated in narrow regions in the less dense network are considered as border, in the denser network only the most peripheral nodes of these regions are classified as borders. This result shows that the method is sensitive to the spatial resolution of the nodes of the geographical network.

3.4. Border detection in a rewired network

In this section, we address the results of the border-detection methodology when applied to rewired versions of geographical networks. Note that, when using rewired versions of a network it is reasonable to consider that part of the properties of the original (non-rewired) network are maintained, i.e. some regions of the rewired networks will still have the properties implied by the physical constraints while others will acquire non-geographical behavior (e.g. emergence of small world effects). Therefore, it is expected that the border-detection methodology applied to these rewired network will not consider as border some of the former border nodes and additionally it will classify as border some of the former internal (non-border) nodes.

In order to evaluate this process, four different rewired versions of the network presented in figure 3(b) were analyzed. Figure 6 shows the obtained networks with 0.1% (a), 1% (b), 5% (c) and 10% (d) of rewired edges. Note that, in this figure, the nodes are already colored according to the border/non-border classification. The threshold criteria used to define the border in all cases was the same as used previously, i.e. the configuration which best approximates the physical border of the original network. That is the reason why many of the detected border nodes still remain near the physical border. Particularly for the first case, (a), the very low percentage of rewiring did not change the properties of the network and, as a consequence, the obtained result is quite similar to the original, non-rewired network. On the other hand, for the three remaining cases, the classification of the nodes as border nodes included some former internal nodes and excluded some former border nodes. Note that this result can also be understood as an extension of the proposed methodology to non-geographical networks.

3.5. Detected borders using other measurements

The results of the border detection obtained by using the node degree, the betweenness centrality and the ASPL are shown in figures 7–9, respectively. For all cases, the best threshold value was chosen in the same way as described previously. Among these three measurements, the best result was found for the node degree. In this case, the border was properly detected at the expense of many internal nodes, which are not border, being classified as border. The same issue occurs, in a worse manner, when the betweenness centrality is used to detect the borders. Finally, when using the ASPL to detect the borders, it is clear from figures 8(a) and (b) that only the nodes located far from the geographical center of the network are classified as border, resulting in a completely incorrect result. In figure 8(c), the detection of the border presented a similar result with respect to the result obtained by using the diversity entropy.

In addition to these three measurements, we also evaluated the diffusion-based algorithm (second eigenvector) proposed by Eriksen et al  [14]. Again, it was not possible to detect the borders, although the algorithm seems to be promising for detecting communities in the graphs.

In order to investigate the relationship between these four measurements and the diversity entropy, table 1 shows their Pearson correlation coefficients. It can be noted from this table that no significant correlation was found with respect to any of the four considered networks, except for the network from figure 3(c) (3D sphere), whose diversity entropy presented high correlation with the ASPL.

3.6. Real-world examples

In order to best illustrate the potential of the proposed methodology, in this section we present results concerning two real-world network examples: the geographical network of the urban streets of São Carlos and the word association network derived from the book Alice's Adventures in Wonderland.

The network of the streets of São Carlos has N = 4537 nodes and E = 7527 edges. The diversity entropy was estimated for h = 1,  ..., 5 using the probabilistic algorithm. In order to determine the borders of this network, the threshold of the diversity entropy was manually chosen. The obtained result is shown in figure 10, where the green nodes represent the border nodes. This figure also shows the railway and the highways that cross the urban perimeter of the city. These structures impose important physical constraints in the planning of the streets of the city, slicing the urban area and giving rise to many border nodes along these ways. Observe that such border nodes, though not being directly related to the periphery of the network, were all properly detected by the proposed diversity methodology.

The second real-world example analyzed, the word association network from the book Alice's Adventures in Wonderland, has N = 1929 nodes and E = 9290 edges. This network was built considering the words as nodes and by linking adjacent words in order to define the edges. To reduce the size of the network, the words were lemmatized (i.e. reduced to their canonical form) and stop-words (e.g. the verb to be, prepositions, conjunctions, pronouns and articles) were not considered. In addition, we excluded nodes of the network which had less than two connections and selected the most connected component of the remaining network. The resulting network, which has N = 505 and E = 1102, is shown in figure 11. The diversity entropy was computed for this network (h = 1,  ..., 5) using the deterministic algorithm and the ten lowest and highest values, together with their corresponding word, are shown in figure 11. It is clear that the most internal words (corresponding to the most internal nodes) seem to correspond to the most important terms in the book, while the border words tend to present a secondary nature regarding the main subjects in this book. In this sense, it is possible that the diversity quantifies in some way the centrality of words and concepts in texts.

Figure 12 shows the Pearson correlation coefficients obtained for the above two real networks considering the diversity entropy and the other three previously introduced measurements: degree, betweenness centrality and ASPL. The obtained results indicate that there is no strong correlation between these measurements.

4. Conclusions

Despite all the current investigations in complex networks research, some important related issues have received little attention. Given that real-world networks are necessarily finite, one important aspect concerns the definition and identification of their borders, a concept immediately related to internality/externality of nodes. The current work has addressed these important problems. More specifically, we used the recently introduced concept of diversity (e.g. [7, 8]) in order to quantify the potential of accessibility from and to each node while considering a specific dynamics, in the present case self-avoiding random walks. Nodes with more balanced transition probabilities to other nodes, expressed in terms of entropy, are understood as being more internal to the network, whereas the other nodes are associated with the network borders.

The concept of borders is immediate and intuitive in geographical networks, where the more internal nodes are usually found at the geometrically more internal regions. Therefore, we give special attention to this type of network in order to motivate and validate our approach. Interestingly, the application of the diversity methodology to geographical networks identified as borders not only the nodes at the peripheral regions, but also those nodes which are geographically more internal but are near geographical discontinuities slicing the network. The definition of borders in non-geographical networks has been an important open question. We showed that it is possible to extend the diversity approach to define and identify borders in this type of network. We also compared the diversity approach with methods founded on other measurements such as degree, betweenness centrality, average shortest path lengths and the second eigenvector. None of these alternative approaches turned out to be able to properly detect the borders, even in the case of geographical networks. We also found no significant correlation between the diversity and those measurements, which further corroborates that that feature does provide additional information about the topology of the networks. The potential of the methodology proposed for the identification of the borders was illustrated with respect to both theoretical (i.e. regular, geographical and knitted networks) and real-world networks (i.e. the urban network of the town of São Carlos and Carroll's Alice's Adventures in Wonderland). Several findings and insights were yielded by the adopted approach, including the impressive performance for identification of the borders in knitted networks, a non-geographical structure. In addition, it was showed that the urban network presents borders not only at its periphery, but also along the railway line and highway that happen to cross that town. In the case of Carroll's work, the most internal words tended to be more immediately related to the main thematic of the book, whereas the most external nodes (borders) related to secondary concepts.

The effectiveness of the reported approach has paved the way to a number of promising further investigations, including its application for the identification of the borders in several important real-world networks such as gene regulation, airports and anatomical networks [17]. A particularly interesting prospect would be to apply the methodology to detect the borders in pictures where the objects do not have well-defined contours (e.g. are composed of textures and clouds).

Acknowledgments

B A N Travençolo is grateful to FAPESP for financial support (07/02938-5), M P Viana thanks FAPESP for financial support (07/50882-9) and L da F Costa thanks CNPq (301303/06-1 and 573583/2008-0) and FAPESP (05/00587-5) for financial support. We also thank Lucas Antiqueira for building the network of words from Carroll's book.

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