Local network connectivity optimization: an evaluation of heuristics applied to complex spatial networks, a transportation case study, and a spatial social network

Formulation of the network connectivity problem

For the description of the optimization methodology the following nomenclature will be used (see Table 1). In connectivity optimization, network nodes are first segmented and assigned to ’close’ and ’distant’ sets by a chosen threshold distance D from the network’s focal node F. The number of nodes ν is a network is N, and nodes are separated into two sets based on their shortest network path distances to the focal node, d(ν, F). The nodes that are within this distance are assigned to the ‘close’ set, N_C ⊂ N, i.e., ν ∈ N_C if d(ν, F) ≤ D and F ∈ N_C (see Fig. 1A). The nodes that are outside the threshold shortest network path distance to the focal node, D, are assigned to the ‘distant’ set, N_D ⊂ N, i.e., ν ∈ N_D if d(ν, F) > D. When a new connection is added to the network, the shortest path distance from each distant node to the focal node is recalculated. If there are any distant nodes that are now within the threshold distance to the focal node they are assigned to the new set $N_{i,j}^C$. For example, if a new connection is established between distant node i and close node j, then $k\in N_{i,j}^C$ if k ∈ N_D and d(k, i) + d(i, j) + d(j, F) ≤ D (Fig. 1C). The benefit of this new connection is B(i, j) and the cost associated with the new connection is C(i, j). The optimal solution is the solution with the greatest benefit, or number of new nodes now within the distance to the focal node which can be expressed as the bi-objective function (1)${\displaystyle \begin{array}{c}\hfill O^{\ast }=max\left(B\left(i,j\right)\right)+min\left(C\left(i,j\right)\right)\\ \hfill \text{s.t.}\sum _i{\nu }_i=1,i\in N_C& \hfill \\ \hfill \sum _j{\nu }_j=1,j\in N_D\end{array}}$ where benefits dominate costs. For example, if B(i, j) = B(m, n) and C(i, j) < C(m, n), then the optimal additional edge is between i and j. For nondominated solutions, we select the solution that minimizes the distance to the so-called ideal point. The ideal point represents the solution that simultaneously maximizes the benefit and minimizes the cost. For the analysis in this paper the formulation of the objective function is as follows. For a new edge between i and j the number of nodes in $N_{i,j}^C$ set is the benefit of this new connection, $B\left(i,j\right)=\left|N_{i,j}^C\right|$, and the cost of the new connection is the length of the edge C(i, j) = d(i, j). Therefore, (2)${\displaystyle \begin{array}{c}\hfill O^{\ast }=max\left(\left|N_{i,j}^C\right|\right)+min\left(d\left(i,j\right)\right)\\ \hfill \text{s.t.}\sum _i{\nu }_i=1,i\in N_C& \hfill \\ \hfill \sum _j{\nu }_j=1,j\in N_D\end{array}}$ Most of the heuristics presented below are dependent on the number of iterations (t) and terminate when the solutions converge, O^t = O^t−1, or the solution does not improve, O^t < O^t−1.

Local search methodology

For a network of size N the number of new connections to evaluate has an upper bound of N²∕4, therefore heuristics may be employed to identify (nearly) optimal solutions quicker than an exhaustive search as networks get larger. These optimization algorithms require a search space to explore and using nodal characteristics we create such a multidimensional solution space (see Table 2). These nodal characteristics are explored to find the critical network properties for connectivity optimization and their impact on the performance in finding the optimal solution. As several nodes in a network can potentially have the same nodal characteristic values the local searches include a random shuffling routine. In detail, the following individual node characteristics were used to create the dimensions of the solution space: (i) distance to the focal node, (ii) degree centrality, (iii) closeness centrality, (iv) betweenness centrality, (v) eigenvector centrality, (vi) pagerank centrality, (vii) weighted clustering coefficient, and(viii) the neighbor nodes directly connected to the node. Nodes were sorted by their distance to the focal node and moving in this solution dimension may result in lower connectivity length costs but may not maximize the number of nodes ultimately connected to the focal node. In contrast, sorting nodes by their centrality, i.e., the importance of the node, could result in maximizing the number of nodes within the specified distance to the focal node but with the possibility of higher connectivity length costs compared to selecting nodes by other characteristics. To avoid these two extreme possibilities, several commonly used measures of centrality are explored: degree, the number of edges incident to a node; closeness centrality, the average length of the shortest path between the node and all other nodes in the network (Bavelas, 1950); betweenness centrality, the frequency of a node included in the shortest paths between all other node pairs (Freeman, 1977); eigenvector centrality, which is a relative sorting of nodes such that nodes with high values are connected to other nodes with high values (Newman, 2008); and pagerank centrality, a variant of eigenvector centrality that sorts nodes based on their probability of being connected to a randomly selected node and which is commonly used in web-page rankings (Brin & Page, 1998). The weighted clustering coefficient of a node is the count of the triplets in the neighborhood of the node and accounts for the weights of the edges times the maximum possible number of triplets that could occur (Barrat et al., 2004). The weights used here are the spatial distances between nodes w_ij = d(i, j). The nodes that were directly connected to the node under evaluation were also used.

Network connectivity optimization heuristics

The following techniques were implemented for the network connectivity optimization study from their extensive use in optimization: hill climbing with random restart (Russell & Norvig, 2004); stochastic hill climbing (Greiner, 1992); hill climbing with a variable neighborhood search (Mladenović & Hansen, 1997); simulated annealing, which has a history of applications in graph problems (Kirkpatrick, Gelatt & Vecchi, 1983; Johnson et al., 1989; Kirkpatrick, 1984); and genetic algorithms, which has been successfully used for combinatorial optimization (Anderson & Ferris, 1994; Jaramillo, Bhadury & Batta, 2002). A Tabu heuristic was not employed as it has been observed to be an inferior method for multi-objective optimization problems compared to simulated annealing and genetic algorithms (Golden & Skiscim, 1986; Kim et al., 2016). Parameter selection was simplified for easy comparison of the methods (see the Supplemental Information for the algorithms). To ensure that the heuristics did not converge on suboptimal solutions due to the initial starting values, random restart, i.e., randomly selecting initial nodes to avoid local optima and running the routine until the optimal solution is found, was used.

Exhaustive search (ES). The exhaustive search calculates the solution for every pair of distant and close nodes for a network (see Algorithm 1 in the Supplemental Information). While this approach ensures that the optimal edges are found, as the number of nodes increases and therefore the number of possible connections between close and distant nodes increases, it can become computationally expensive and timely to implement. Since the results of ES provide the optimal solution, the times it takes to find evaluate all solutions are used to benchmark the other heuristics.

Hill climbing (HC). The solution space was observed to be hilly from the exhaustive search results, so several modifications were introduced to the hill climbing technique to avoid getting stuck in suboptimal solutions (Algorithm 2 in the Supplemental Information). A stochastic hill climbing (HCS), an advanced search method based on HC, routine is also explored where the selection of nodes for the next iteration is randomly picked with (3)${\displaystyle \text{probability}\left(i,j\right)=\frac{O\left(i,j\right)}{\sum _{\left(m,n\right)}O\left(m,n\right)},}$ which terminates when an improved solution is no longer found (Algorithm 3 in the Supplemental Information). A hill climbing algorithm is coupled with a variable neighborhood (HCVN) where the size of the neighborhood starts with the nearest neighbors (η = 1) and is updated as follows: (4)${\displaystyle \eta =\left\{\begin{array}{cc}1\hfill & \text{if}{O}^t>O^{t-1}\hfill \\ \eta +1\hfill & \text{if}{O}^t\le O^{t-1}\hfill \end{array}\right.,}$ and the HCVN method terminates after n_max is reached (Algorithm 4 in the Supplemental Information).

Simulated annealing (SA). As a meta-heuristic approach, the simulated annealing method randomly selects an initial solution from the solution space to avoid entrapment in a local optima. At each iteration, the heuristic evaluates the neighboring solutions and if it does not find an improved solution, it moves to a new solution with the following probability: (5)${\displaystyle \text{probability}\left(i,j\right)=exp\left(-\frac{O^{t-1}-O\left(i,j\right)}{t}\right).}$ The distance of the move decreases with the number of iterations until a better solution is no longer found (Algorithm 5 in the Supplemental Information).

Genetic algorithm (GA). The genetic algorithm begins with a population of P randomly selected solutions with a set of chromosomes composed of genes which represent the weights of selecting a neighbor and are all initialized to unity (Algorithm 6 in the Supplemental Information). During each iteration of the method, solution scores (fitnesses) are computed by (6)${\displaystyle f\left(i,j\right)=\frac{O\left(i,j\right)}{\sum _{\left(m,n\right)}O\left(m,n\right)},}$ and a new generation of solutions are selected based on the following probability condition (7)${\displaystyle \text{probability}\left(i,j\right)=\frac{s\ast f\left(i,j\right)+\left(1-s\right)}{\sum _{\left(m,n\right)}\left(s\ast f\left(m,n\right)+\left(1-s\right)\right)},}$ where s is the selection coefficient. Weak selection, s ≪ 1, is used to ensure that random mutations impact solution frequency. Crossover is conducted by alternating the weights for the offspring from each parent, also known as cycle crossover (Oliver, Smith & Holland, 1987). Mutations are introduced at a low rate μ ≪ 1 for each gene and increase the nodal characteristic selection weight by one. The probability that characteristic m is used to find a neighbor for node i is given by (8)${\displaystyle \text{probability(characteristic)}=\frac{\text{gene}\left(i,m\right)}{\sum _k\text{gene}\left(i,k\right)∕K},}$ where K is the total number of nodal characteristics. This formulation ensures that the nodal characteristics that improve the solution increase in weight which results in a greater probability they will be selected for neighborhood exploration, and ultimately reduces the size of the neighborhood search. Among these methods, the genetic algorithm presented here introduces a novel chromosome formulation where the genes are not properties of a specific variable but weights for the probability to move in a given direction in the solution space. This allows the method to occasionally explore different nodal characteristics (at mutation rate μ) while conducting a local neighborhood search.

Simulated data: random networks

To test the efficacy of these optimization heuristics in finding the optimal new network connections they were applied to randomly generated networks that vary in complexity and size. Several types of random graph networks were generated to analyze the efficacy of the optimization heuristics for systems with different topologies which are generally representative of naturally occurring and built systems: (1) Erdös-Rényi networks, (2) Watts-Strogatz networks, (3) Barabási and Albert networks, (4) Klemm and Eguílez networks, (5) Delaunay triangulation networks, and (6) Voronoi diagrams. Erdös-Rényi random networks are constructed by randomly creating connection between pairs of nodes with a probability (Erdös & Rényi, 1959). These networks, even though they have random connections, consistently have short average path lengths and irregular connections, both of which are well found in natural systems. The Watts-Strogatz networks also have random connections but the networks also form clusters, another feature commonly found in real-world networks (Watts & Strogatz, 1998). The Barabási-Albert model produces random structures with a small number of highly connected nodes, ’hubs’, which are observed in numerous types of networks (Barabási & Albert, 1999; Albert & Barabási, 2002). Klemm and Eguílez networks have random connections, clusters, and hubs (Klemm & Eguílez, 2002). See Figs. S.1A–S.1D and Prettejohn, Berryman & McDonnell (2011) for the algorithms used to generate these networks.

We also introduce two novel types of random planar network versions of the Voronoi diagram and the Delaunay triangulation (Figs. S.1E and S.1F). Planarity is particularly important in many fields and networks generated from Voronoi diagrams and Delaunay triangles have been used in spatial health epidemiology (Johnson, 2007), transportation flow problems (Steffen & Seyfried, 2010; Pablo-Martì & Sánchez, 2017), terrain surface modeling (Floriani, Falcidieno & Pienovi, 1985), telecommunications (Meguerdichian et al., 2001), computer networks design (Liebeherr & Nahas, 2001), and hazard avoidance systems in autonomous vehicles (Anderson, Karumanchi & Iagnemma, 2012). Delaunay triangulation maximizes the minimum angles between three nodes to generate planar graphs with consistent network characteristics while Voronoi diagrams, the dual of a Delaunay triangulation, are composed of points and cells such that each cell is closer to its point than any other point (Delaunay, 1934). To modify these edges are removed from network nodes randomly based on their distance from the focal node with probability (9)${\displaystyle p_R\cdot \frac{max\left(d\left(i,F\right),d\left(j,F\right)\right)}{\underset{k}{max}d\left(k,F\right)},}$ where p_R is the removal probability and weighted by the normalized edge distance from the focal node. When edges are randomly removed from the connected Delaunay network or Voronoi network, with weights given by node distance from a focal node, these networks display some of the properties similarly found in the networks mentioned above, such as complexity and randomness. Yet, these networks have the added component of being planar and having edge weights that can be framed as physical distances.

To compare the efficacy of different optimization methods for different network topologies, identifying the best set of parameters are critical. Parameter values were selected for each type of random network to ensure network complexity (Table S.6 summarizes the parameters which were used in the analysis). Variation in network size was also explored and the most connected node in each network was selected as the focal node. Uniformly randomly generated edge weights in [0,1] were used for the network distances and the threshold distance was set to ensure that half of the nodes were initially within the distance to the focal node. The costs and benefits were normalized using the ranges from the exhaustive search routine as a benchmark to compare the results from the different optimization methods.

Empirical data: transportation case study

To complement the random network analysis, the network connectivity optimization methods were applied to a study of urban transportation planning. Network connectivity optimization methods were used to evaluate the potential costs and benefits of increased thoroughfare connectivity for student walking or biking to school. It is assumed that expanding the connectivity around a school would allow for more households, and students, to be included within the walking distance to the school. If more students actively commute to school, this reduces the busing costs for the school system and increases the health and academic achievement of the students (Centers for Disease Control and Prevention, 2010). Yet, streets have associated land, construction, and maintenance costs that are primarily dependent on their length.

Networks composed of street edges and residence nodes around several schools from a representative US school system were used for the analysis. Ten suburban and rural schools from Knox County, TN, were selected for the analysis, including seven elementary and three middle, that would benefit the most from increased thoroughfare connectivity, i.e., had the most students within the Euclidean walking distance but not the network distance to the school (characteristics of these schools are provided in the Table S.3). Urban schools were not used since the street connectivity around the schools was significantly high and the from additional thoroughfares would be low. Residential parcels and street networks were provided by the Knoxville-Knox County Metropolitan Planning Commission and the residential parcels were converted to nodes and placed on the nearest street edge. The residences within 1 mile and 1.5 miles, for the elementary schools and the middle schools respectively, are considered close nodes while the nodes outside of these distances were classified as distant nodes (see Fig. 2). The school networks do not generally display the characteristics of complex network: they had low average degree, large path lengths, and were not efficient, yet they did have power distributions of connectivity with few intersections having a large number of street connections (see Supplemental Information Table S.3). The networks were evaluated with each optimization method to maximize the number or close residences connected to the school and minimize the distance of the new thoroughfare. The costs and benefits of these street connections were normalized using the ranges from the exhaustive search routine as a benchmark to compare the results from the different optimization methods.

Empirical data: social network analysis

The topology of a network and the management of its system can improve information flow and have been shown to help counter the negative effects of social and environmental crises (Helbing et al., 2015). Specifically, increased social network connectivity can reduce the time information spreads among its members. For example, quickly notifying those nearby and unaware of an active shooter event can save lives. Using a set of social network data coupled with location data, we evaluated the heuristics to optimize social network connectivity for those nearby a specific location.

The data set used for this analysis was from Gowalla, a location-based social networking website where users share their locations by checking-in, and this data set was collected and published by Cho, Myers & Leskovec (2011). This social network is undirected and consists of 196,591 nodes (members) and 950,327 edges (social connections). There is a total of 6,442,890 check-ins of these members over the period of February 2009 to October 2010 (network characteristics are provided in the Supplemental Information Table S.4). For the analysis we simulated 100 crisis incidents at a highly populated urban place, Grand Central Terminal in New York City (NY). Grand Central Terminal was selected as the location of the simulated events as it is a major transportation hub located in the center of the city that serves over a million commuters and visitors daily. New York City is also a major metropolis with tens of thousands of Gowalla users present in the data set and the city has a history of incidents, such as terror attacks. Ten dates were selected at random and for each date ten times were randomly selected between 1200 and 1800 local to simulate a crisis event (see Fig. 3).

The social network problem was formulated such that an event occurred at the location (Grand Central Station) and members who were within 0.5-miles of the event should be notified. The members within Grand Central Station are automatically notified of the event whereas those outside can be notified through their online social network if a member in their network is aware. The event is considered to be serious enough that those members aware of it will share the information on the social network. New connections are evaluated based on their number of additional nearby members who become aware of the event.