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Discovering Important Nodes Through Graph Entropy Encoded in Urban Space Syntax

Potentially influential spaces in the spatial networks of cities can be detected by means of the entropy participation ratios. Local (connectivity) and global (centrality) entropies are considered. While the connectivity entropy has a tendency to increase with the city size, the centrality entropy is decreasing that reflects the global connectedness of cities. In urban networks, the local and global properties of nodes are positively correlated that indicates the intelligibility of cities. Correlations between entropy participation ratios can be used in purpose of intelligibility measurements and city networks comparisons.


Movement, Intelligibility, and Navigation
As a city grows through the accumulation of new buildings and areas, its street network emerges which links all open spaces together and creating new spaces in the expanding settlement, [1].
Within the equal socio-economic frameworks and physical constrains, a human moves in a direction that provides him or her the potential for possible further movement [2]. J. Gibson calls such interaction between humans and environments natural vision [3]. The natural vision is a combination of visual factors affecting behavior. The next step has been made in by the theory of natural movement that suggests that movement within a spatial network linking the buildings will be determined by the grid configuration itself [4]. A key result of space syntax research is that the pattern of spatial integration in the urban grid is a key determinant of pedestrian movement in cities across the world, [7].
This degree of correlation between aggregate human movement rates and spatial configuration is surprising since the analysis of dual city graphs incorporate neither many of the factors considered important in previous models of human behavior in urban environments (such as the motivations and the origin-destination information) nor direct account was taken of the metric properties of space [8]. Nevertheless, the robustness of agreement between global integration (centrality) and rush hour movement rates is now supported by a number of similar studies of pedestrian movement in different parts of the world and in an everyday commercial work of the Space Syntax Ltd., [9]. Similar results also exist for vehicular movement [10] showing that the spatial configuration of the urban grid is in itself a consistent factor in determining movement flows. B.Hillier and his colleagues [4] have shown that the majority of human-pedestrian movement occurs along lines of sight, and that the more integrated (in terms of connection to other lines of sight) a line is, the more movement exists along it. A research has established that pedestrian movement is more impacted by the number of turns than by distance travelled. Streets from which other streets can be reached with fewer direction changes attract much more people [11].
Land uses which seek movement, such as markets and retail, then naturally migrate towards higher movement locations, while others, perhaps residential, prefer low movement locations [5]. The emerging structure of the spatial pattern gives rise to a natural movement pattern, making some spaces higher in co-presence than others. Because more integrated streets attract more people, they also tend to attract retail and other land uses that depend upon the volumes of pedestrian traffic, and consequently the volumes of both pedestrians and uses are multiplied [11]. Economic growth in the highly integrated streets feeds back onto the structure of the grid improving its inter-accessibility. This process will often stabilize at a certain level related to the original grid properties that generated the natural movement in the first place [1].
In the present paper, we investigate the spatial networks (in the dual representation) of several compact urban patterns (Sec. 2). Potentially influential spaces can be detected in the urban texture by means of their contributions into the entropies (by the entropy participation ratio) of the dual graph representation of a city (Sec. 3).
The key observation is that for any graph one can introduce two distinct classes of entropies. The first class is related to a local property of nodes in the graph (connectivity). Entropies of another class are calculated with respect to the different centrality measures quantifying the global property of nodes in the graph. Local (connectivity) and global (centrality) entropies are quite different. For instance, while the connectivity entropy has a tendency to increase with the city size, the centrality entropy is decreasing, since a big city usually has "broadways", the itineraries of prominent centrality connecting separated districts of the city (Sec. 3).
In urban networks, the local and global properties of nodes are positively correlated that indicates the intelligibility of cities (Sec. 4). We show that correlations between entropy participation ratios can be used in purpose of intelligibility measurements and city networks comparisons (Sec. 4).

Spatial networks of compact urban patterns
A spatial network of a city is a network of spatial elements that constitute the urban environment. They are derived from maps of open spaces (streets, places, and roundabouts). Open spaces may be broken down into components; most simply, these might be street segments, which can be linked into a network via their intersections and analyzed as a networks of movement choices. The study of spatial configuration is instrumental in predicting human behavior, for instance, pedestrian movements in urban environments [12]. A set of theories and techniques for the analysis of spatial configurations is called space syntax [13]. Although, in its initial form, space syntax was focused mainly on patterns of pedestrian movement in cities, later the various space syntax measures of urban configuration had been found to be correlated with the different aspects of social life, [14].
Open spaces are all interconnected, so that one can travel within open spaces from everywhere to everywhere else. It is sometime difficult to decide what should be an appropriate spatial element of the complex space involving large number of open areas and many interconnected paths. Decomposition of a space map into a complete set of intersecting axial lines, the fewest and longest lines of sight that pass through every open space comprising any system, produces an axial map or an overlapping convex map respectively. In Fig. 1, we have presented the axial map drawn for the small city of Rothenburg ob der Tauber (Bavaria, Germany). Axial lines and convex spaces may be treated as the spatial elements (nodes of a morphological graph), while either the junctions of axial lines or the overlaps of convex spaces may be considered as the edges linking spatial elements into a single graph unveiling the topological relationships between all open elements of the urban space. In what follows, we shall call this morphological representation of urban network as a dual graph. An example of such a morphological representation for the axial map of the city of Rothenburg ( Fig. 1) is displayed on Fig. 2. We have studied five different compact urban patterns. Two of them are situated on islands: Manhattan (with an almost regular grid-like city plan) and the network of Venice canals (imprinting the joined effect of natural, political, and economical factors acting on the network during many centuries). In the old city center of Venice that stretches across 122 small islands in the marshy Venetian Lagoon along the Adriatic Sea in northeast Italy, the canals serve the function of roads.
We have also considered two organic cities founded shortly after the Crusades and developed within the medieval fortresses: Rothenburg ob der Tauber, the medieval Bavarian city preserving its original structure from the 13 th century, and the downtown of Bielefeld (Altstadt Bielefeld), an economic and cultural center of Eastern Westphalia.
To supplement the study of urban canal networks, we have investigated that one in the city of Amsterdam. Although it is not actually isolated from the national canal network, it is binding to the delta of the Amstel river, forming a dense canal web exhibiting a high degree of radial symmetry.
The scarcity of physical space is among the most important factors determining the structure of compact urban patterns. Some characteristics of studied dual city graphs are given in Tab.1. There, N is the number of open spaces (streets/canals and places) in the urban pattern (the number of nodes in the dual graphs), M is the number of junctions (the number of edges in the dual graphs); the graph diameter, diam(G) is the maximal depth (i.e., the graph-theoretical distance) between two vertices in a dual graph; the intelligibility parameter (see Sec. 4) estimates navigability of the city, suitability for the passage through it.
3 Graph entropy encoded in urban space syntax A major task of space syntax analysis in so far is the discovery of potentially influential spaces in the urban texture which can attract a high volume of movement. It is also important to rank the commercially potent spaces in a city with regard to the rest.
To address such a challenge, we use an information theoretic model that combines information theory with statistical techniques.
The entropy parameter that is a measure of the uncertainty associated with a random variable helps to identify the most important nodes or a set of such nodes in a large dual city graph G(V, E).
Let P be the probability distribution on the vertex set V (G), so that each node i ∈ V is characterized by the probability p i ∈]0, 1] such that N i=1 p i = 1. Within space syntax, the probability distribution can be defined with relevance to any space syntax measure discussed in the previous subsections. To be certain, we consider just two examples of such the distributions.
We define the probability distribution π related to the local connectivity measure quantified by the degree of a vertex v ∈ V is the number of edges that end at v: The connectivity is a local measures that shows how well an open space is intersecting with other spaces in the urban pattern, in which M = |E| is the total number of junctions in the city. It is worth to mention that given random walks defined on the connected undirected graph G, such that a walker moves in one step to another node randomly chosen among all its nearest neighbors, then the probability distribution (2) coincides with the unique stationary distribution of random walkers. The most important property of the stationary distribution is that if G is not bipartite, then the distribution of any node i in random walks tends to its stationary value π i as t → ∞.
Another probability distribution we consider is the global choice measure [15] quantifying the relative structural importance of the node in the graph. A space i ∈ V has a strong choice value when many of the shortest paths, connecting all spaces to all spaces of a system, passes through it. Any probability distribution P leads us directly to the entropy (or the structural information content) of a graph: with the standard convention that 0 ·log 2 (1/0) = 0. The information entropy as defined in (4) had been introduced by C.E. Shannon [19]. We shall refer to the entropy calculated with regard to the probability distribution (2) as the connectivity entropy and the entropy calculated with respect to (3) as the centrality entropy. If all nodes of the graph are characterized by the equal probability p i with respect to the probability distribution P , the relevant entropy should be maximal, H max = N log 2 N. For instance, both entropies calculated in regard the probability distributions (2) and (3) exhibit the maximum value for regular graphs, in which all nodes are characterized by equal centrality and connectivity. Alternatively, the entropy (4) tends to zero as all p i → 0, i = k, but one p k → 1.
The minimal values of both entropies are achieved for a star graph. In Fig. 3, we have plotted the values of the both entropies calculated for the five compact urban patterns ordered with regard to the dual graph sizes. It is then apparent that for the small organic German medieval cities and for the canal network of Amsterdam pertaining to a high degree of central symmetry the typical values of both entropies are proximate. However, the entropies demonstrate the alternative tendencies for much larger urban networks. The values of centrality entropy evidently decrease for the larger dual city graphs indicating the presence of a few nodes providing the essentially strong choice. Alternatively, the values of connectivity entropy blow up for larger networks bespeaking that the connectivity probability distributions are smoothed due to the excessive growth of the number of junctions between streets in large cities.
The function H(G, P ) is continuous with respect to changing the value of one of the probabilities p i , symmetric being unchanged if open spaces are re-ordered, and additive being independent of how the network is regarded as being divided into parts. The last property allows to compute the entropy of a graph as a sum of partial entropies pertinent to its subgraphs and even single nodes. In such a context, the interpretation of important nodes are those who have the most effect of the graph entropy. The importance of a node in terms of its contribution into entropy with regard to the probability distribution P can be estimated by the entropy participation ratio (EPR) defined as A key observation, relevant to all compact urban patterns we have studied is that the prominent nodes contributing conspicuously into the centrality entropy of the graph may be inferior with respect to the connectivity entropy and vice versa. In order to exemplify that, we present in Fig. 4   data in (Fig. 4) showed considerable variations, it is obvious that the both entropies follow a general tendency that can be quantified by means of a correlation coefficient indicating the strength and direction of a relationship between two data sets. For instance, Pearson's product-moment linear correlation coefficient [20] between the PRE calculated with respect to the centrality and connectivity entropies for the streets in the downtown of Bielefeld amounts to 0.7734. Let us recall that the correlation is 1 in the case of an increasing linear relationship between data sets. Proximate values of linear correlation coefficient can also be found between the PRE data for the streets in other dual city graphs (see the diagram in Fig. 5). Local and global properties of nodes come along in urban spatial networks.
The notion that local and global aspects of urban structures are related is at the foundation of space syntax. In particular, it has been suggested in [8] that if cities act as mechanisms for generating contact between local inhabitants and strangers, then a spatial mechanism at the basis of this would be likely to include correlations between local and global movement structures.

Discussion and Conclusion: Intelligibility and Navigation
In space syntax, correlations between local property of a space (connectivity) and global configurational variables (integration) constitute a measure of the intelligibility, the global parameter quantifying the part-whole relationship within the spatial configuration. Intelligibility describes how far the depth of a space from the street layout as a whole can be inferred from the number of its direct connections [15] that is most important to way-finding and perception of environments [16,17]. More integrated areas were also found to be more "legible" by the residents who perceived their "neighborhood" to be of a greater size, [18].
Natural movement relies on an adequate level of intelligibility which has been found to encourage peoples way-finding abilities. Spatial integration was found to be correlated with observed movement, with the more intelligible area showing stronger correlations, [18].
If there is practically no relationship between how connected a node is and how integrated it is with the overall structure, the rela-tionship between space and movement is weak and the environment seems to be confusing. While being in a such "unintelligible" layout, people get lost more frequently and change directions often that makes navigation more difficult [21]. Movement behavior of humans in highly complex and unintelligible urban areas like the multi-level urban complexes has been investigated in [22,23] (the citation appears in [8]). The key findings of the research were that even in highly unintelligible areas movement was largely predictable from aspects of the environment. However, whereas in intelligible urban areas the single variable of integration accounted for the substantial proportion of variance in movement flows, in the multi-level complexes a much wider range of variables needed to be taken into account.
In the traditional space syntax approach, the strong area definition and good intelligibility are identified in an intelligibility scattergram and then by means of the Visibility Graph Analysis (VGA) [5]. In statistics, a scatter plot is a useful summary of a set of two variables, usually drawn before working out a linear correlation coefficient or fitting a regression line. Each node of the dual graph contributes one point to the scatter plot. The resulting pattern indicates the type and strength of the relationship between the two variables, and aids the interpretation of the correlation coefficient [6].
The measure of spatial integration for each node i ∈ G is usually taken the mean distance (called mean depth in space syntax [15]), however the analysis varies for specific case studies, and the precise measures of the graph are chosen to best correlate.
We prefer to use the global choice parameter (3) as a measure of spatial integration. The scatter plot for the downtown of Bielefeld (in the log-log scale) which shows the relationship between connectivity and global choice (centrality) is sketched on Fig. 6. The pattern of dots (representing the certain open spaces in the downtown of Bielefeld) slopes from lower left to upper right that suggests a positive correlation between the connectivity and centrality variables being studied. A line of best fit computed using the method of linear regression exhibits the slope 0.253. Let us note that the value of Pearson's coefficient of linear correlations between the data samples of connectivity and global choice for Bielefeld equals to 0.681.
We have demonstrated that the correlations between local and global properties within the spatial configurations of urban networks (intelligibility) can be quantified by regarding at least three different methods. In the previous subsection, we have estimated it by means of Pearson's coefficient of linear correlations [20] between the EPR indices calculated with regard to the centrality and connectivity entropies. The level of correlations can also be reckoned by the slope of the regression line fitting the data of the scatter plot drawn in the logarithmic scale. Eventually, we can directly compute the correlation coefficient between the uniformly ordered connectivity and integration values. In order to show the compatibility of all three methods, we collect the results of all intelligibility estimations for the five compact urban patterns that we studied in one diagram (see Fig. 5). It is clear from the diagram (5) that being an important characteristic related to a perception of place and navigation within that, intelligibility can be used in a purpose of comparison between urban networks. The obvious advantage of intelligibility is that it does not depend upon the network size.
The intelligibility indices estimated by means of Pearson's coefficient of linear correlations between the EPR indices have been given in Tab. 1.

Acknowledgment
The work has been supported by the Volkswagen Foundation (Germany) in the framework of the project: "Network formation rules, random set graphs and generalized epidemic processes" (Contract no Az.: I/82 418). The authors acknowledge the multiple fruitful discussions with the participants of the workshop Madeira Math Encounters XXXIII, August 2007, CCM -CENTRO DE CIÊNCIAS MATEMÁTICAS, Funchal, Madeira (Portugal).