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Optimization of synchronizability in complex spatial networks
Accepted Manuscript Optimization of synchronizability in complex spatial networks Nameer Al Khafaf, Mahdi Jalili
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S0378-4371(18)31135-X https://doi.org/10.1016/j.physa.2018.09.030 PHYSA 20091
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Physica A
Received date : 16 March 2018 Revised date : 15 July 2018 Please cite this article as: N.A. Khafaf, M. Jalili, Optimization of synchronizability in complex spatial networks, Physica A (2018), https://doi.org/10.1016/j.physa.2018.09.030 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
A model is proposed to construct random spatial networks. A rewiring-based method is proposed to improve synchronizability of the networks. Optimized networks have increased local and global efficiency.
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*Manuscript Click here to view linked References
Optimization of Synchronizability in Complex Spatial Networks Nameer Al Khafaf and Mahdi Jalili School of Engineering, RMIT University, Melbourne, Australia Email for correspondence: [email protected]
Abstract Many real-world phenomena can be modelled as spatial networks where nodes have distinct geographical location. Examples include power grids, transportation networks and the Internet. This paper focuses on optimizing the synchronizability of spatial networks. We consider the eigenratio of the Laplacian Matrix of the connection graph as a metric measuring the synchronizability of the network and develop an efficient rewiring mechanism to optimize the topology of the network for synchronizability, i.e., minimizing the eigenratio. The Euclidean distance between two connected nodes is considered as their connection weights, and the sum of all connection weights is defined as the network cost. The proposed optimization algorithm constructs spatial networks with a certain number of nodes and a predefined network cost. We also study the topological properties of the optimized networks. This algorithm can be used to construct spatial networks with optimal synchronization properties.
Keywords Optimization, Synchronization, Spatial Networks, Power Grids
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1 . Introduction Many natural and human-made phenomena can be modelled as networks where a number of entities (known as nodes, agents or vertices) are connected through interactions (known as links or edges) [1]. Examples of such systems include the Internet, human brain, social networks and critical networks such as energy grids, transportation and water distribution networks. Interacting dynamical networks, i.e., those with dynamical systems sitting on the nodes, might exhibit collective behaviour such as synchronization, consensus and coherence [2-7]. When two or more dynamical systems interact over a networked structure and their mutual coupling strength is strong enough, their trajectories might get into synchrony resulting in coherent motion. Synchronization has potential applications in science and engineering and has been subject to heavy investigations. A key question in studying complex networked structures is “what is the interplay between dynamics and network structure?” Structure of a network has a major role in determining its dynamical properties such as synchronizability [8-10]. Network synchronizability is defined as the ease by which the network synchronizes behaviour of individual units sitting on its edges. There are a number of interpretations (and metrics) for network synchronizability[11, 12]. The most widely-used synchronizability measure is the one based on the master stability function approach [13], which results in a pure graph-based metric, i.e., the eigenratio of the Laplacian matrix of the connection graph. It is an effective measure for weighted undirected networks, which can be used as a criterion to enhance network synchronization properties through engineering its structure, e.g., rewiring, link weighting or assigning proper directions to the links [14-16]. Previous studies in networks dynamical properties have been mostly based on considering networks without any spatial tags on the nodes. However, many real systems are indeed spatial networks. A wide range of social, engineering and natural systems can be modelled as spatially distributed networked structures [17]. Transportation networks are prototypic example of spatial networks, where nodes are stations (or junctions) and edges are physical connections (e.g., roads) between them [18, 19]. Networks formed by bus and subway routes, railway links and urban commuters networks are also spatial in nature [20, 21]. Power grids and water supply networks – which have evolved for a rather long time in many countries – are important infrastructures in modern societies. These infrastructures are classified as spatial networks where the nodes have a geographic tag [22-24]. These networks are critical infrastructures for any country and their resiliency is of high importance for governing bodies. Spatial networks exhibit properties, which makes them distinct from other types of network. For example, their nodes have abnormal betweenness centrality [25]. Due to geographical constraints, degree distribution of spatial networks does not include high degrees and is usually peaked. Spatial constraints 2
imply that the tendency of making links is towards short-range connection, resulting in a flat behaviour in assortativity and high clustering coefficient. It has also been frequently shown that correlation between topology, traffic and distance is nonlinear in spatial networks [17]. Spatial constraints of networks have distinctive consequence on the processes happening on them such as synchronization [2] and searching [26]. Space constrains often lead to an increased level of interdependence between critical networks, where failures in one network can have drastic influence on others [27]. A recent study showed that small failures in one of the electricity suppliers in Italy in September 2003 caused shutdown of a power station leading to failure in some internet nodes, which in turn caused breakdowns in many other power stations [27-29]. This makes it important to control such interdependent spatial networks against cascaded failures. In this work, we study synchronizability of spatial networks with fixed network connection cost and propose an optimization procedure to construct such networks with optimal levels of synchronizability. The networks are spatial andd the nodes have fixed geographical location. Also, the networks have fixed connection cost that is defined as the sum of all link weights. Link weight is defined as the Euclidean distance between the end nodes. This is a realistic representation of many real spatial networks such as power grids, transportation and water distribution networks, where creating a physical link between any two nodes incorporate cost such that the longer the link, the higher the cost. We introduce a model to construct seed networks with predefined size and network cost. The seed networks are then optimized employing efficient links rewiring and an optimization algorithm. Our simulations show that the proposed guided rewiring strategy is effective in improving the synchronizability of the network. We also study structural properties of the optimized networks.
2 . Random Spatial Graphs with Fixed Connection Cost There are many network generation models in the literature; however there is no model to construct spatial networks with connection cost. Here we propose a novel generation model to construct such networks. Spatial networks can be modelled using graph theory in Euclidean Space. Let’s consider a network with N nodes and E edges. Each node represents the Euclidean coordinates for an object whereas each edge represents the link or connection between two nodes in the Euclidean space. The power grid is an example of a spatial network, where the generating station, transformer and load represent the nodes having distinct location in space and the overhead lines or underground cables represent the links between the nodes [22, 24, 29]. Transportation network is another application for spatial network where a node represents the train station, tram or bus stop whereas the edge represents the railway between stations or the journey line between stops. Fig.1 shows an example of a power grid network modelled as a spatial network. As it is evident from Fig. 1b, each node has a unique coordinate which defines its geographical location. The edges represent transmission lines which are directly related to the distance. The distance is then 3
related to the power cable which represent the cost for establishing one link between the
and
node.
Indeed, the connection cost of a link equals the Euclidean distance between its end nodes. The sum of all the individual link weights represents the total cost for creating the network. Network cost usually plays a major role in decision making as such minimizing cost has always been an objective for management executives. Spatial networks can be weighted or unweighted. A node i is said to be adjacent to node j when there exists a direct connection between them. In our model of a weighted spatial network, the weight represents the Euclidean distance between node i and node j if the connection exists. All links in a spatial network between the
and
node can be summarized in the adjacency matrix A = [aij], which is a
square
matrix whose entries are defined by:
(1) where (2) where (xi,yi) are the coordinates of node i in x- and y-axis. The number of edges and their weights connected to each node is denoted by the degree matrix
= [dij] which is a diagonal matrix defined as:
(3) The Laplacian Matrix is defined as: (4) or (5)
In this work, we aim at constructing a network with N nodes and a predefined connection cost C such that it has high levels of synchronizability. First, we construct an initial seed network, and then use an iterative optimization algorithm to optimize the synchronizability. The initial network is constructed by randomly distributing
nodes in Euclidean space where each node has a unique coordinate stored in a two-
dimensional matrix (Fig. 2a). Then, the distance matrix is calculated as:
(6) 4
To construct the initial network, first the nodes are chosen randomly and get connected to their nearest neighbours (a nearest neighbour of a node is the one with the least Euclidean distance from that node). When all nodes have at least one connection, one obtains M island (Fig 2b). In order to connect two islands, we start from a randomly chosen island and connect it to its nearest island. This process continues until a connected network is obtained (Fig. 2c). The final stage is to allocate the remaining cost randomly across the network. To this end, all the link weights (i.e., distances between the end nodes) are summed and subtracted from available cost C. The remaining value is the total link weight that is randomly distributed in the network (Fig. 2d). The final network is a connected network with N nodes and cost C. The next step is to rewire the links to optimize the synchronizability of the network.
3 . Network Synchronizability Synchronization is the most well-known collective behaviour observed in dynamical networks [2]. When two or more dynamical systems interact and their coupling is stronger than a certain threshold, they can get into synchrony and oscillate in pace [30]. Synchronization has many applications in science and engineering and has been heavily investigated in the last two decades. Let’s consider the random spatial networks constructed using the above algorithm. Let’s suppose that a system sits on the nodes. These dynamical systems interact through the network connections. The equations of the motion of the dynamical network read
(7) where
are n-dimensional state vectors, g(.) represent the dynamics of (identical) individual systems
and α is the coupling strength.
is the Laplacian matrix of the connection graph and
is the
projection matrix indicating how the individual dynamical systems are coupled. An important question for studying Eq. (7) is how to determine the synchronizing coupling strength for which the dynamical systems are synchronized. One can argue that the individual dynamical systems are globally synchronized if starting from random initial conditions, their trajectories converges as time goes to infinity. When the trajectories converge, if the initial conditions should be close enough to one another, the system is termed to be locally synchronized. Several techniques have been proposed to study the behaviour of the above dynamical equations. For example, connection graph stability method obtains a sufficient
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condition for the stability of the synchronized solution, that is if the coupling strength is larger than the threshold predicted by this method, the synchronized solution is globally asymptotically stable[31, 32]. Master stability function method is an alternative approach to study local stability of the synchronized solution, providing a necessary condition for the synchronization[13]. The master stability function approach is based on the variational equations of Eq. (7) that studies the Lyapunov exponents to check the stability. The largest Lyapunov exponent of the variational equations is denoted as the master stability function, which accounts for linear stability of the synchronized solution, that is, to provide local synchrony the coupling strength must be such that the master stability function is negative [13]. The synchronization condition obtained in this way has two parts; one coming from the individual dynamical systems and another from the connection graph. It has been shown that the eigenratio of the Laplacian matrix, i.e., the largest eigenvalue divided by the second smallest one, represents the degree of synchronization in the network such that the smaller the eigenratio, the better the synchronizability of the network[13]. The eigenratio is defined as (8) where
and
are the largest and the second smallest eigenvalue of the Laplacian matrix, respectively.
Note that since the Laplacian matrix is a zero row-sum with positive diagonal entries, it is a semi-positive matrix with all eigenvalues being non-negative. The smallest eigenvalue is zero, and
is non-zero for
connected networks. The eigenratio of a graph is the most frequently used metric measuring its synchronizability [15, 33-35]. In this work we optimize the structure of spatial networks (constructed under limited network cost) for synchronizability as measured by the eigenratio.
4 . Optimization Procedure Since the eigenratio R is used as a measure for network synchronizability, the network topology reflected in the Laplacian matrix will have the sole impact on the synchronizability of the network, and thus in the optimization procedure. The main aim of the optimization algorithm is to minimize the eigenratio and optimize the objective function: (9) (10)
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The optimization procedure engineers the network structure by structural changes in wirings to minimize the eigenratio. The connection links are rewired through a guided rewiring process, keeping the network cost fixed at C. In order to do the rewiring, at each step of the optimization algorithm, one of the edges is chosen for disconnection based on a criterion and one or more edges with the same cost, i.e., the same total weight as the disconnected edge, are created. This way the network cost, i.e., total summation of connection weights, is kept unchanged during the optimization process. The optimization algorithm is as follows. First, the eigenratio and the strength values of the nodes in the initial spatial network is calculated. Strength of a node is defined as the summation of all weighted links connected to that node. The optimization procedure is based on guided link rewiring that is to disconnect an edge and create one or more edges with the same connection cost as the disconnected one. To select an edge to disconnect, we choose a node with a probability proportional to its strength; the higher the strength of a node the higher the probability of choosing that node. We limit this to those nodes with the strength higher than the average strength of all nodes. As the candidate node is identified, the link with the largest weight pointing to this node is selected and disconnected. The next step is to create one or more links with weights equal to the disconnected link. To this end, a node is chosen among those with strength lower than the average, such that the lower is the strength of a node; the higher is the probability of its selection. As this node is identified, an edge with weights less or equal to the disconnected link is created between this node and a randomly chosen node. If the weight of a newly created link is less than the disconnected node, the remaining weight is randomly distributed between the nodes. This way the connection cost of the original and the rewired network remains the same. As the rewiring process is completed, the eigenratio of the new network is obtained. If the eigenratio of the new network is less than the old one, the rewiring is accepted. Otherwise, we accept the rewiring with a very small probability. The optimization process stops when steady state solution is obtained and the eigenratio does not change over a certain number of iterations. The steps below summarize the optimization algorithm: 1) Calculate the average edge weight of all nodes in the network; 2) Calculate the probability of choosing a node from set of nodes having edge weight greater than the average; 3) If the set of nodes having edge weight greater than the average is selected: a. Select a random node from the set of selected nodes; b. Disconnect a random edge from the selected node and add it to the budget; c. Select a node having the lowest edge weight for rewiring;
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d. Rewire a randomly selected edge to the selected node based on the remaining budget such that the weight of the disconnected edge is equivalent to connected ones. 4) Otherwise: a. Select a node with the highest edge weight from the set of nodes having edge weight less than the average; b. Disconnect the edge with the lowest weight from the selected nodes and add it to the budget; c. Select a random node from the set of selected nodes; d. Rewire a random edge to the selected node based on the remaining budget such that the weight of the disconnected edge is equivalent to connected ones. 5) Calculate the eigenratio of the network; 6) Accept the rewiring if the new eigenratio is less than the old one, otherwise the rewiring is rejected or accepted with small probability. 7) The process stops if the eigenratio does not change over specified iteration number or when the iteration number reaches a predetermined number (40,000 iterations in this work). The modelling and the simulation environment of the optimization algorithm have two parameters that can be altered to get structures leading to optimal synchronizability. The first parameter is the probability for selecting a random node or rewiring a random edge. This is done by assigning higher probabilities for selecting certain nodes or rewiring specific edges compared to others. The other parameter is the connection cost which can be altered to see how the network synchronizability changes as the connection cost varies. Fig. 3a shows a sample network with N = 500 nodes and cost C = 34,624. The final optimized network is shown in Fig. 3b. In the next section, we examine the effectiveness of the proposed optimization procedure and study the structural properties of the optimized network.
5 . Results We test the above optimization algorithm on networks with N = 500 (C = 31,375), N = 1000 (C = 85,236). First, the initial spatial network is constructed with the mentioned algorithm, and then the proposed rewiring-based optimization procedure is performed. Fig. 4 shows the eigenratio R of the networks as a function of the iterations of the optimization processes. It is seen that the proposed rewiringbased optimization procedure is effective in decreasing R, and thus improving the synchronizability of the network. The algorithm exponentially decreases the eigenratio in the early stages, and then fine tunes the topology of the network as the optimization proceeds. The optimization process is more effective in larger 8
networks; the proposed optimization process decreases the eigenratio of networks with N = 1000 (N = 500) by about nine (four) times. We next investigate the topological properties of the optimized network. To this end, we study the structural properties including assortativity, mean node betweenness centrality and global and local efficiency. Global efficiency of a network indicates the global communicability between the nodes and is proportional to inverse of the average path length, and is defined as [36]: 1 N N 1
Eg
1 , qij
i, j
(11)
where qij is the length of the shortest path between the nodes i and j. The local efficiency of node i is defined as [36]: 1 Ni Ni
El ,i
1
1 , Gi qij
i, j
(12)
where Ni is the number of neighbours of i and Gi is the subgraph of its neighbours excluding i. The local efficiency of the network is obtained by making average over all the nodes, as: El
1 N
El ,i ,
(13)
i
In order to take into account the centrality of network nodes, a frequently used metric is betweenness centrality, which measures the significance of the nodes in the shortest paths[37]. Node-betweenness centrality (load or traffic) ρi of node iis defined as i
pu i i p u
,
(14)
pu
where Γpu is the number of shortest paths between nodes p and u in the graph; and Γpu(i) is the number of these shortest paths making use of node i. Often the average betweenness centrality of the nodes is studied in networks. A frequently used metric for resiliency of networks is assortativity, indicating the degree-degree correlation[38], i.e. assortativity of the network. In assortative networks, nodes with high (low) degrees tend to connect to those with high (low) degrees, whereas in disassortative networks, nodes with high (low) degree tend to be linked to those with low (high) degree. In order to calculate the degree correlation, one may use the Pearson correlation of the degrees at both ends of the edges of the network 2
1 M
r
ki k j aij j i
1 M
j i
1 ki 2
k j aij 2
1 M
j i
1 2 ki 2
k 2j aij
1 M
j i
1 ki 2
,
(15)
k j aij
where M is the total number of the edges of the network and ki is the degree of node i. If r> 0, the network 9
is assortative, whereas r< 0 indicates a disassortative network. For r = 0 there is no correlation between the node-degrees. The assortativity is calculated on unweighted version of the network, while other measures are calculated on the original weighted networks. Fig. 5 and 6 shows assortativity, average betweenness centrality, global and local efficiency measures for networks with N = 500 and N = 1000, respectively. The pattern of the evolution of these structural properties is almost similar for networks of different sizes. The optimized networks are less assortative than the original spatial networks. This indicates that the tendency of high (low) degree nodes to be connected to high (low) degree nodes decreases in the optimized networks. The average betweenness centrality also decreases in the optimized networks, indicating that the nodes are less loaded (i.e., fewer shortest paths passing through) in the optimized networks than the original networks. Interestingly, as the optimization proceeds, both global and local efficiency measures increase. This indicates that when the networks are optimized to have high levels of synchronizability (i.e., minimum R), both their global and local connectedness improves. This means that the nodes have better global communicability in the optimized networks; the length of the shortest paths becomes shorter in average. Furthermore, the communicability of the local neighbourhood of the nodes is also facilitated (i.e., the length of the shortest paths between the nodes in the local neighbourhood decreases). Our main application in this manuscript is power grids, where the nodes (generators, loads and transformers) have fixed locations, and the links represent wirings between them. In order to show the applicability of the proposed rewiring-based optimization procedure on power grids, we consider a real network representing Electricity Transmission Network of Victorian state of Australia[39]. This network has N= 69 nodes and cost C = 6,532,900 (Fig. 7a). Fig.7b shows the final optimized networks where the location of nodes is kept unchanged. It is seen from Fig. 7c that the proposed optimization algorithm is effective in reducing the eigenratio, and thus enhancing the network synchronizability. It is worth mentioning that the metric considered here as a measure of synchronizability, the eigenratio of the Laplacian matrix, is not valid for directed networks or those composed of non-identical oscillators. In addition to that, the eigenratio metric is only valid when the inner linking matrix is positive definite or positive semi-definite. This limits the application of the proposed rewiring-based optimization algorithm for weighted undirected networks and for networks with general inner linking matrix. One can however use similar metrics for directed networks; a possible synchronizability measure for directed networks is the ratio of the real part of the largest eigenvalue divided by the real part of the second smallest eigenvalue of the Laplacian matrix.
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6 . Conclusion In this paper we proposed an algorithm to construct highly synchronizable weighted undirected spatial networks with a certain number of nodes and network cost. The network cost is defined as the summation of weights of all edges, where the weight of an edge is obtained as the Euclidean distance between its two end nodes. In order to construct spatial networks with optimal synchronizability, we first constructed seed networks with the given size and cost. Then, we used an efficient rewiring strategy to optimize the eigenratio of the Laplacian matrix of the network, as a measure of its synchronizability. We proposed a technique to disconnect an edge and create one or more edges with the same connection cost, thus preserving the network cost during the optimization process. Our numerical simulations on sample networks showed that the proposed rewiring-based optimization effectively reduced the eigenratio, and thus improved the synchronizability. We also monitored how the structural properties of networks evolved during the optimization process. The optimized networks showed less assortativity, less average betweenness centrality, and higher global and local efficiency, as compared to the initial seed networks. The proposed optimization strategy can be efficiently used to construct highly synchronizable weighted undirected spatial networks, with potential application in designing more efficient power grids networks.
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Acknowledgments Mahdi Jalili is supported by Australian Research Council through project No. DP170102303.
Figures:
Figure 1: a) A sample grid showing how electricity is distributed and regulated with the coordinates showing the location of power plant, substation and consumers. b) It represents the spatial network model of (a). Triangles represent power plants, rectangles represent substation, circles represent consumers and lines represent high voltage transmission lines.
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Figure 2: Generation of a connected network with N nodes and having the minimum connection cost. a) 100 nodes are distributed in space randomly having distinct x and y coordinate. b) m islands are established after each node is connected to the nearest neighbour node. c) 100 nodes spatial network established using the minimum cost such that the network is connected and each node is able to reach every other node in the graph. d) The initial spatial network as an input for optimization with N nodes and cost C
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Figure 3: Effect of optimization algorithm on the network structure. (a) example network with N = 500 and C = 34,624 as initial network for the optimization procedure. (b) The final optimized network.
Figure 4: The eigenratio R as a function of iterations for networks with a) N = 500 and C = 31,375, and b) N = 1000 and C= 85,236. The proposed rewiring-based optimization procedure gradually decreases R, and thus improved the synchronizability. The number of nodes N and total network cost C remain unchanged during the optimization process. The graphs show averages over 40,000 realisations.
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Figure 5: Structural properties of networks with N = 500 and C = 31,375 as a function of iteration steps of the proposed optimization procedure. The networks are those used in Fig. 4a. The structural properties include a) assortativity, b) mean node betweenness centrality, c) global efficiency and d) local efficiency.
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Figure 6: Structural properties of networks with N = 1,000 and C = 85,236 as a function of iteration steps of the proposed optimisation procedure. The structural properties include a) assortativity, b) mean node betweenness centrality, c) global efficiency and d) local efficiency.
Figure 7: (a) Victoria’s Electricity Transmission Network with N = 69 and C = 6,532,900 as initial network for the optimization procedure, (b) the final optimized network, and c) the eigenratio as a function of the iterations.
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PII: DOI: Reference:
S0378-4371(18)31135-X https://doi.org/10.1016/j.physa.2018.09.030 PHYSA 20091
To appear in:
Physica A
Received date : 16 March 2018 Revised date : 15 July 2018 Please cite this article as: N.A. Khafaf, M. Jalili, Optimization of synchronizability in complex spatial networks, Physica A (2018), https://doi.org/10.1016/j.physa.2018.09.030 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
A model is proposed to construct random spatial networks. A rewiring-based method is proposed to improve synchronizability of the networks. Optimized networks have increased local and global efficiency.
1
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Optimization of Synchronizability in Complex Spatial Networks Nameer Al Khafaf and Mahdi Jalili School of Engineering, RMIT University, Melbourne, Australia Email for correspondence: [email protected]
Abstract Many real-world phenomena can be modelled as spatial networks where nodes have distinct geographical location. Examples include power grids, transportation networks and the Internet. This paper focuses on optimizing the synchronizability of spatial networks. We consider the eigenratio of the Laplacian Matrix of the connection graph as a metric measuring the synchronizability of the network and develop an efficient rewiring mechanism to optimize the topology of the network for synchronizability, i.e., minimizing the eigenratio. The Euclidean distance between two connected nodes is considered as their connection weights, and the sum of all connection weights is defined as the network cost. The proposed optimization algorithm constructs spatial networks with a certain number of nodes and a predefined network cost. We also study the topological properties of the optimized networks. This algorithm can be used to construct spatial networks with optimal synchronization properties.
Keywords Optimization, Synchronization, Spatial Networks, Power Grids
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1 . Introduction Many natural and human-made phenomena can be modelled as networks where a number of entities (known as nodes, agents or vertices) are connected through interactions (known as links or edges) [1]. Examples of such systems include the Internet, human brain, social networks and critical networks such as energy grids, transportation and water distribution networks. Interacting dynamical networks, i.e., those with dynamical systems sitting on the nodes, might exhibit collective behaviour such as synchronization, consensus and coherence [2-7]. When two or more dynamical systems interact over a networked structure and their mutual coupling strength is strong enough, their trajectories might get into synchrony resulting in coherent motion. Synchronization has potential applications in science and engineering and has been subject to heavy investigations. A key question in studying complex networked structures is “what is the interplay between dynamics and network structure?” Structure of a network has a major role in determining its dynamical properties such as synchronizability [8-10]. Network synchronizability is defined as the ease by which the network synchronizes behaviour of individual units sitting on its edges. There are a number of interpretations (and metrics) for network synchronizability[11, 12]. The most widely-used synchronizability measure is the one based on the master stability function approach [13], which results in a pure graph-based metric, i.e., the eigenratio of the Laplacian matrix of the connection graph. It is an effective measure for weighted undirected networks, which can be used as a criterion to enhance network synchronization properties through engineering its structure, e.g., rewiring, link weighting or assigning proper directions to the links [14-16]. Previous studies in networks dynamical properties have been mostly based on considering networks without any spatial tags on the nodes. However, many real systems are indeed spatial networks. A wide range of social, engineering and natural systems can be modelled as spatially distributed networked structures [17]. Transportation networks are prototypic example of spatial networks, where nodes are stations (or junctions) and edges are physical connections (e.g., roads) between them [18, 19]. Networks formed by bus and subway routes, railway links and urban commuters networks are also spatial in nature [20, 21]. Power grids and water supply networks – which have evolved for a rather long time in many countries – are important infrastructures in modern societies. These infrastructures are classified as spatial networks where the nodes have a geographic tag [22-24]. These networks are critical infrastructures for any country and their resiliency is of high importance for governing bodies. Spatial networks exhibit properties, which makes them distinct from other types of network. For example, their nodes have abnormal betweenness centrality [25]. Due to geographical constraints, degree distribution of spatial networks does not include high degrees and is usually peaked. Spatial constraints 2
imply that the tendency of making links is towards short-range connection, resulting in a flat behaviour in assortativity and high clustering coefficient. It has also been frequently shown that correlation between topology, traffic and distance is nonlinear in spatial networks [17]. Spatial constraints of networks have distinctive consequence on the processes happening on them such as synchronization [2] and searching [26]. Space constrains often lead to an increased level of interdependence between critical networks, where failures in one network can have drastic influence on others [27]. A recent study showed that small failures in one of the electricity suppliers in Italy in September 2003 caused shutdown of a power station leading to failure in some internet nodes, which in turn caused breakdowns in many other power stations [27-29]. This makes it important to control such interdependent spatial networks against cascaded failures. In this work, we study synchronizability of spatial networks with fixed network connection cost and propose an optimization procedure to construct such networks with optimal levels of synchronizability. The networks are spatial andd the nodes have fixed geographical location. Also, the networks have fixed connection cost that is defined as the sum of all link weights. Link weight is defined as the Euclidean distance between the end nodes. This is a realistic representation of many real spatial networks such as power grids, transportation and water distribution networks, where creating a physical link between any two nodes incorporate cost such that the longer the link, the higher the cost. We introduce a model to construct seed networks with predefined size and network cost. The seed networks are then optimized employing efficient links rewiring and an optimization algorithm. Our simulations show that the proposed guided rewiring strategy is effective in improving the synchronizability of the network. We also study structural properties of the optimized networks.
2 . Random Spatial Graphs with Fixed Connection Cost There are many network generation models in the literature; however there is no model to construct spatial networks with connection cost. Here we propose a novel generation model to construct such networks. Spatial networks can be modelled using graph theory in Euclidean Space. Let’s consider a network with N nodes and E edges. Each node represents the Euclidean coordinates for an object whereas each edge represents the link or connection between two nodes in the Euclidean space. The power grid is an example of a spatial network, where the generating station, transformer and load represent the nodes having distinct location in space and the overhead lines or underground cables represent the links between the nodes [22, 24, 29]. Transportation network is another application for spatial network where a node represents the train station, tram or bus stop whereas the edge represents the railway between stations or the journey line between stops. Fig.1 shows an example of a power grid network modelled as a spatial network. As it is evident from Fig. 1b, each node has a unique coordinate which defines its geographical location. The edges represent transmission lines which are directly related to the distance. The distance is then 3
related to the power cable which represent the cost for establishing one link between the
and
node.
Indeed, the connection cost of a link equals the Euclidean distance between its end nodes. The sum of all the individual link weights represents the total cost for creating the network. Network cost usually plays a major role in decision making as such minimizing cost has always been an objective for management executives. Spatial networks can be weighted or unweighted. A node i is said to be adjacent to node j when there exists a direct connection between them. In our model of a weighted spatial network, the weight represents the Euclidean distance between node i and node j if the connection exists. All links in a spatial network between the
and
node can be summarized in the adjacency matrix A = [aij], which is a
square
matrix whose entries are defined by:
(1) where (2) where (xi,yi) are the coordinates of node i in x- and y-axis. The number of edges and their weights connected to each node is denoted by the degree matrix
= [dij] which is a diagonal matrix defined as:
(3) The Laplacian Matrix is defined as: (4) or (5)
In this work, we aim at constructing a network with N nodes and a predefined connection cost C such that it has high levels of synchronizability. First, we construct an initial seed network, and then use an iterative optimization algorithm to optimize the synchronizability. The initial network is constructed by randomly distributing
nodes in Euclidean space where each node has a unique coordinate stored in a two-
dimensional matrix (Fig. 2a). Then, the distance matrix is calculated as:
(6) 4
To construct the initial network, first the nodes are chosen randomly and get connected to their nearest neighbours (a nearest neighbour of a node is the one with the least Euclidean distance from that node). When all nodes have at least one connection, one obtains M island (Fig 2b). In order to connect two islands, we start from a randomly chosen island and connect it to its nearest island. This process continues until a connected network is obtained (Fig. 2c). The final stage is to allocate the remaining cost randomly across the network. To this end, all the link weights (i.e., distances between the end nodes) are summed and subtracted from available cost C. The remaining value is the total link weight that is randomly distributed in the network (Fig. 2d). The final network is a connected network with N nodes and cost C. The next step is to rewire the links to optimize the synchronizability of the network.
3 . Network Synchronizability Synchronization is the most well-known collective behaviour observed in dynamical networks [2]. When two or more dynamical systems interact and their coupling is stronger than a certain threshold, they can get into synchrony and oscillate in pace [30]. Synchronization has many applications in science and engineering and has been heavily investigated in the last two decades. Let’s consider the random spatial networks constructed using the above algorithm. Let’s suppose that a system sits on the nodes. These dynamical systems interact through the network connections. The equations of the motion of the dynamical network read
(7) where
are n-dimensional state vectors, g(.) represent the dynamics of (identical) individual systems
and α is the coupling strength.
is the Laplacian matrix of the connection graph and
is the
projection matrix indicating how the individual dynamical systems are coupled. An important question for studying Eq. (7) is how to determine the synchronizing coupling strength for which the dynamical systems are synchronized. One can argue that the individual dynamical systems are globally synchronized if starting from random initial conditions, their trajectories converges as time goes to infinity. When the trajectories converge, if the initial conditions should be close enough to one another, the system is termed to be locally synchronized. Several techniques have been proposed to study the behaviour of the above dynamical equations. For example, connection graph stability method obtains a sufficient
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condition for the stability of the synchronized solution, that is if the coupling strength is larger than the threshold predicted by this method, the synchronized solution is globally asymptotically stable[31, 32]. Master stability function method is an alternative approach to study local stability of the synchronized solution, providing a necessary condition for the synchronization[13]. The master stability function approach is based on the variational equations of Eq. (7) that studies the Lyapunov exponents to check the stability. The largest Lyapunov exponent of the variational equations is denoted as the master stability function, which accounts for linear stability of the synchronized solution, that is, to provide local synchrony the coupling strength must be such that the master stability function is negative [13]. The synchronization condition obtained in this way has two parts; one coming from the individual dynamical systems and another from the connection graph. It has been shown that the eigenratio of the Laplacian matrix, i.e., the largest eigenvalue divided by the second smallest one, represents the degree of synchronization in the network such that the smaller the eigenratio, the better the synchronizability of the network[13]. The eigenratio is defined as (8) where
and
are the largest and the second smallest eigenvalue of the Laplacian matrix, respectively.
Note that since the Laplacian matrix is a zero row-sum with positive diagonal entries, it is a semi-positive matrix with all eigenvalues being non-negative. The smallest eigenvalue is zero, and
is non-zero for
connected networks. The eigenratio of a graph is the most frequently used metric measuring its synchronizability [15, 33-35]. In this work we optimize the structure of spatial networks (constructed under limited network cost) for synchronizability as measured by the eigenratio.
4 . Optimization Procedure Since the eigenratio R is used as a measure for network synchronizability, the network topology reflected in the Laplacian matrix will have the sole impact on the synchronizability of the network, and thus in the optimization procedure. The main aim of the optimization algorithm is to minimize the eigenratio and optimize the objective function: (9) (10)
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The optimization procedure engineers the network structure by structural changes in wirings to minimize the eigenratio. The connection links are rewired through a guided rewiring process, keeping the network cost fixed at C. In order to do the rewiring, at each step of the optimization algorithm, one of the edges is chosen for disconnection based on a criterion and one or more edges with the same cost, i.e., the same total weight as the disconnected edge, are created. This way the network cost, i.e., total summation of connection weights, is kept unchanged during the optimization process. The optimization algorithm is as follows. First, the eigenratio and the strength values of the nodes in the initial spatial network is calculated. Strength of a node is defined as the summation of all weighted links connected to that node. The optimization procedure is based on guided link rewiring that is to disconnect an edge and create one or more edges with the same connection cost as the disconnected one. To select an edge to disconnect, we choose a node with a probability proportional to its strength; the higher the strength of a node the higher the probability of choosing that node. We limit this to those nodes with the strength higher than the average strength of all nodes. As the candidate node is identified, the link with the largest weight pointing to this node is selected and disconnected. The next step is to create one or more links with weights equal to the disconnected link. To this end, a node is chosen among those with strength lower than the average, such that the lower is the strength of a node; the higher is the probability of its selection. As this node is identified, an edge with weights less or equal to the disconnected link is created between this node and a randomly chosen node. If the weight of a newly created link is less than the disconnected node, the remaining weight is randomly distributed between the nodes. This way the connection cost of the original and the rewired network remains the same. As the rewiring process is completed, the eigenratio of the new network is obtained. If the eigenratio of the new network is less than the old one, the rewiring is accepted. Otherwise, we accept the rewiring with a very small probability. The optimization process stops when steady state solution is obtained and the eigenratio does not change over a certain number of iterations. The steps below summarize the optimization algorithm: 1) Calculate the average edge weight of all nodes in the network; 2) Calculate the probability of choosing a node from set of nodes having edge weight greater than the average; 3) If the set of nodes having edge weight greater than the average is selected: a. Select a random node from the set of selected nodes; b. Disconnect a random edge from the selected node and add it to the budget; c. Select a node having the lowest edge weight for rewiring;
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d. Rewire a randomly selected edge to the selected node based on the remaining budget such that the weight of the disconnected edge is equivalent to connected ones. 4) Otherwise: a. Select a node with the highest edge weight from the set of nodes having edge weight less than the average; b. Disconnect the edge with the lowest weight from the selected nodes and add it to the budget; c. Select a random node from the set of selected nodes; d. Rewire a random edge to the selected node based on the remaining budget such that the weight of the disconnected edge is equivalent to connected ones. 5) Calculate the eigenratio of the network; 6) Accept the rewiring if the new eigenratio is less than the old one, otherwise the rewiring is rejected or accepted with small probability. 7) The process stops if the eigenratio does not change over specified iteration number or when the iteration number reaches a predetermined number (40,000 iterations in this work). The modelling and the simulation environment of the optimization algorithm have two parameters that can be altered to get structures leading to optimal synchronizability. The first parameter is the probability for selecting a random node or rewiring a random edge. This is done by assigning higher probabilities for selecting certain nodes or rewiring specific edges compared to others. The other parameter is the connection cost which can be altered to see how the network synchronizability changes as the connection cost varies. Fig. 3a shows a sample network with N = 500 nodes and cost C = 34,624. The final optimized network is shown in Fig. 3b. In the next section, we examine the effectiveness of the proposed optimization procedure and study the structural properties of the optimized network.
5 . Results We test the above optimization algorithm on networks with N = 500 (C = 31,375), N = 1000 (C = 85,236). First, the initial spatial network is constructed with the mentioned algorithm, and then the proposed rewiring-based optimization procedure is performed. Fig. 4 shows the eigenratio R of the networks as a function of the iterations of the optimization processes. It is seen that the proposed rewiringbased optimization procedure is effective in decreasing R, and thus improving the synchronizability of the network. The algorithm exponentially decreases the eigenratio in the early stages, and then fine tunes the topology of the network as the optimization proceeds. The optimization process is more effective in larger 8
networks; the proposed optimization process decreases the eigenratio of networks with N = 1000 (N = 500) by about nine (four) times. We next investigate the topological properties of the optimized network. To this end, we study the structural properties including assortativity, mean node betweenness centrality and global and local efficiency. Global efficiency of a network indicates the global communicability between the nodes and is proportional to inverse of the average path length, and is defined as [36]: 1 N N 1
Eg
1 , qij
i, j
(11)
where qij is the length of the shortest path between the nodes i and j. The local efficiency of node i is defined as [36]: 1 Ni Ni
El ,i
1
1 , Gi qij
i, j
(12)
where Ni is the number of neighbours of i and Gi is the subgraph of its neighbours excluding i. The local efficiency of the network is obtained by making average over all the nodes, as: El
1 N
El ,i ,
(13)
i
In order to take into account the centrality of network nodes, a frequently used metric is betweenness centrality, which measures the significance of the nodes in the shortest paths[37]. Node-betweenness centrality (load or traffic) ρi of node iis defined as i
pu i i p u
,
(14)
pu
where Γpu is the number of shortest paths between nodes p and u in the graph; and Γpu(i) is the number of these shortest paths making use of node i. Often the average betweenness centrality of the nodes is studied in networks. A frequently used metric for resiliency of networks is assortativity, indicating the degree-degree correlation[38], i.e. assortativity of the network. In assortative networks, nodes with high (low) degrees tend to connect to those with high (low) degrees, whereas in disassortative networks, nodes with high (low) degree tend to be linked to those with low (high) degree. In order to calculate the degree correlation, one may use the Pearson correlation of the degrees at both ends of the edges of the network 2
1 M
r
ki k j aij j i
1 M
j i
1 ki 2
k j aij 2
1 M
j i
1 2 ki 2
k 2j aij
1 M
j i
1 ki 2
,
(15)
k j aij
where M is the total number of the edges of the network and ki is the degree of node i. If r> 0, the network 9
is assortative, whereas r< 0 indicates a disassortative network. For r = 0 there is no correlation between the node-degrees. The assortativity is calculated on unweighted version of the network, while other measures are calculated on the original weighted networks. Fig. 5 and 6 shows assortativity, average betweenness centrality, global and local efficiency measures for networks with N = 500 and N = 1000, respectively. The pattern of the evolution of these structural properties is almost similar for networks of different sizes. The optimized networks are less assortative than the original spatial networks. This indicates that the tendency of high (low) degree nodes to be connected to high (low) degree nodes decreases in the optimized networks. The average betweenness centrality also decreases in the optimized networks, indicating that the nodes are less loaded (i.e., fewer shortest paths passing through) in the optimized networks than the original networks. Interestingly, as the optimization proceeds, both global and local efficiency measures increase. This indicates that when the networks are optimized to have high levels of synchronizability (i.e., minimum R), both their global and local connectedness improves. This means that the nodes have better global communicability in the optimized networks; the length of the shortest paths becomes shorter in average. Furthermore, the communicability of the local neighbourhood of the nodes is also facilitated (i.e., the length of the shortest paths between the nodes in the local neighbourhood decreases). Our main application in this manuscript is power grids, where the nodes (generators, loads and transformers) have fixed locations, and the links represent wirings between them. In order to show the applicability of the proposed rewiring-based optimization procedure on power grids, we consider a real network representing Electricity Transmission Network of Victorian state of Australia[39]. This network has N= 69 nodes and cost C = 6,532,900 (Fig. 7a). Fig.7b shows the final optimized networks where the location of nodes is kept unchanged. It is seen from Fig. 7c that the proposed optimization algorithm is effective in reducing the eigenratio, and thus enhancing the network synchronizability. It is worth mentioning that the metric considered here as a measure of synchronizability, the eigenratio of the Laplacian matrix, is not valid for directed networks or those composed of non-identical oscillators. In addition to that, the eigenratio metric is only valid when the inner linking matrix is positive definite or positive semi-definite. This limits the application of the proposed rewiring-based optimization algorithm for weighted undirected networks and for networks with general inner linking matrix. One can however use similar metrics for directed networks; a possible synchronizability measure for directed networks is the ratio of the real part of the largest eigenvalue divided by the real part of the second smallest eigenvalue of the Laplacian matrix.
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6 . Conclusion In this paper we proposed an algorithm to construct highly synchronizable weighted undirected spatial networks with a certain number of nodes and network cost. The network cost is defined as the summation of weights of all edges, where the weight of an edge is obtained as the Euclidean distance between its two end nodes. In order to construct spatial networks with optimal synchronizability, we first constructed seed networks with the given size and cost. Then, we used an efficient rewiring strategy to optimize the eigenratio of the Laplacian matrix of the network, as a measure of its synchronizability. We proposed a technique to disconnect an edge and create one or more edges with the same connection cost, thus preserving the network cost during the optimization process. Our numerical simulations on sample networks showed that the proposed rewiring-based optimization effectively reduced the eigenratio, and thus improved the synchronizability. We also monitored how the structural properties of networks evolved during the optimization process. The optimized networks showed less assortativity, less average betweenness centrality, and higher global and local efficiency, as compared to the initial seed networks. The proposed optimization strategy can be efficiently used to construct highly synchronizable weighted undirected spatial networks, with potential application in designing more efficient power grids networks.
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Acknowledgments Mahdi Jalili is supported by Australian Research Council through project No. DP170102303.
Figures:
Figure 1: a) A sample grid showing how electricity is distributed and regulated with the coordinates showing the location of power plant, substation and consumers. b) It represents the spatial network model of (a). Triangles represent power plants, rectangles represent substation, circles represent consumers and lines represent high voltage transmission lines.
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Figure 2: Generation of a connected network with N nodes and having the minimum connection cost. a) 100 nodes are distributed in space randomly having distinct x and y coordinate. b) m islands are established after each node is connected to the nearest neighbour node. c) 100 nodes spatial network established using the minimum cost such that the network is connected and each node is able to reach every other node in the graph. d) The initial spatial network as an input for optimization with N nodes and cost C
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Figure 3: Effect of optimization algorithm on the network structure. (a) example network with N = 500 and C = 34,624 as initial network for the optimization procedure. (b) The final optimized network.
Figure 4: The eigenratio R as a function of iterations for networks with a) N = 500 and C = 31,375, and b) N = 1000 and C= 85,236. The proposed rewiring-based optimization procedure gradually decreases R, and thus improved the synchronizability. The number of nodes N and total network cost C remain unchanged during the optimization process. The graphs show averages over 40,000 realisations.
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Figure 5: Structural properties of networks with N = 500 and C = 31,375 as a function of iteration steps of the proposed optimization procedure. The networks are those used in Fig. 4a. The structural properties include a) assortativity, b) mean node betweenness centrality, c) global efficiency and d) local efficiency.
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Figure 6: Structural properties of networks with N = 1,000 and C = 85,236 as a function of iteration steps of the proposed optimisation procedure. The structural properties include a) assortativity, b) mean node betweenness centrality, c) global efficiency and d) local efficiency.
Figure 7: (a) Victoria’s Electricity Transmission Network with N = 69 and C = 6,532,900 as initial network for the optimization procedure, (b) the final optimized network, and c) the eigenratio as a function of the iterations.
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