Optimal transport in multilayer networks for traffic flow optimization

Having given the broader context for our work, we now highlight the main features of transport on multilayer networks. The presence of edges between layers (inter-layer edges) makes a multilayer network fundamentally distinct from a standard single-layer one, as these edges allow passengers to switch between transportation modes. However, this is not the only difference. In fact, in a multilayer network, the various layers have different characteristics. The main one is that the type of transportation cost varies across layers. For example, the cost to build and maintain the infrastructure differs depending on the transportation mode, with subway or rail tracks costing more than a road network. Moreover, the cost assigned to traffic congestion is also different, as road networks are more sensitive to traffic bottlenecks than rail ones. In addition, also the power dissipated differs depending on the means of transportation, as running a tram generally produces less $CO_{2}$ emissions than running a bus. All these different features impact the results of an optimal transport problem, as the network features contributing to the cost function to be optimized vary with layers, and thus also the optimal solution.

Finally, the network topologies themselves vary with layers halu2014emergence, as a bus network has many edges with short lengths, while a rail network tends to have fewer edges but longer. In addition, the weights assigned to each edge differ based on the layer, which can induce a coupling between layers morris2012transport.

In general, a multilayer network is represented as a graph $G(\left\{\mathcal{V}_{\alpha}\right\}_{\alpha},\left\{\mathcal{E}_{\alpha}\right\}_{\alpha},\left\{\mathcal{E}_{\alpha\gamma}\right\}_{\alpha,\gamma})$, where $\mathcal{V}_{\alpha}$ and $\mathcal{E}_{\alpha}$ are the set of nodes and edges in layer $\alpha$, respectively, and $\mathcal{E}_{\alpha\gamma}$ is the set of edges between nodes in layer $\alpha$ and nodes in layer $\gamma$. Here $\alpha=1,\dots,L$ and $L$ is the number of layers. We denote with $N_{\alpha}=|\mathcal{V}_{\alpha}|$ the number of nodes in layer $\alpha$, and with $E_{\alpha}=|\mathcal{E}_{\alpha}|$ the number of edges in layer $\alpha$, $E_{\alpha\gamma}=|\mathcal{E}_{\alpha\gamma}|$ is the number of edges between nodes in layer $\alpha$ and $\gamma$. Finally, we denote with $\mathcal{V}_{0}=\cup_{\alpha}\mathcal{V}_{\alpha}$ the total set of nodes, $\mathcal{E}_{0}=\left(\cup_{\alpha}\mathcal{E}_{\alpha}\right)\,\cup\left(\cup_{\alpha\gamma}\mathcal{E}_{\alpha\gamma}\right)$ the total set of edges, and $N_{0}=|\mathcal{V}_{0}|$ and $E_{0}=|\mathcal{E}_{0}|$ their cardinalities. We assume that edges have lengths $l_{e}>0$, which determine the cost to travel through them.

Transportation networks are relevant examples of this type of structures, where nodes are stations, edges are connections between stations and layers are transportation modes, for instance rails or bus routes. A convenient way to represent multilayer network is with two tensors de2013mathematical: i) an intra-layer adjacency tensor $A$ with entries $A_{uv}^{\alpha}=1$ if there is an edge between nodes $u$ and $v$ in layer $\alpha$, and 0 otherwise. We refer to this type of edges as intra-layer edges; ii) an inter-layer adjacency tensor $\hat{A}$ with entries $\hat{A}_{uv}^{\alpha\gamma}=1$ if there is an edge between node $u$ in layer $\alpha$ and node $v$ in layer $\gamma$, and 0 otherwise. Without loss of generality, in our applications we have $\hat{A}_{uv}^{\alpha\gamma}=0$ if $u\neq v$, meaning that different layers are connected solely by shared nodes. We refer to edges connecting nodes in different layers as inter-layer edges. In the case of transportation networks, the main application studied here, a station could have a bus stop, a train platform and a subway entrance, which allows passengers to switch between communication modes within the same station. For example, one can think of an inter-layer edge as the stairs connecting the subway entrance with the entrance to the train station. Typically, inter-layer edges are thus much shorter than intra-layer edges.

In case of multilayer networks, we need to be careful on how stations connecting multiple transportation modes are represented. In fact, if an entry station connects more than one layer, we may not be able to distinguish in what layer a passenger enters. In other words, if a node $u$ belongs to more than one layer, i.e. a node $u_{\alpha}$ exists for more than one value of $\alpha$, we may not be able to tell whether the passengers entering $u$ entered from $u_{\alpha}$, $u_{\gamma}$ or from any of the other instances of node $u$ in the various layers. To alleviate this problem, we build auxiliary super nodes $u$, which do not belong to any layer in particular but instead connect the various instances of the same node in the various layers together. Specifically, we remove all the inter-layer edges $(u_{\alpha},u_{\gamma})$ and replace them with auxiliary inter-super edges $(u_{\alpha},u)$ connecting all the instances $u_{\alpha}$ of node $u$ with the super node $u$, as in a star graph, so that the original edge $(u_{\alpha},u_{\gamma})$ has been replaced by a 2-edge path $\left\{(u_{\alpha},u),(u_{\gamma},u)\right\}$.

This auxiliary structure allows the model to allocate in an optimal way passengers along the inter-super edges when they enter from a station with connections to more than one layer, thus avoiding to select arbitrary entrances a priori. This becomes relevant in applications where the cost to travel along inter-layer edges is non trivial. For instance, in situations where changing connection impacts the comfort of the passengers.

Moreover, the introduction of super nodes and edges facilitates how we represent the multilayer network. In fact, by adding these auxiliary super nodes and inter-super edges, we only need to consider an individual network adjacency matrix $A$, instead of two separate tensors. This matrix has entries $A_{uv}=1$ if an edge exists between nodes $u$ and $v$ and 0 otherwise, where a node $u$ can be a node $u_{\alpha}$ in layer $\alpha$ or a super node $u$. The set of nodes will then be $\mathcal{V}=\mathcal{V}_{0}\cup\mathcal{V}_{super}$, where $\mathcal{V}_{super}$ is the set of super nodes, and $|\mathcal{V}_{super}|=N_{super}$ is their number, which corresponds to the number of nodes that belong to more than one layer. Similarly, the new set of edges is $\mathcal{E}=\left(\cup_{\alpha}\mathcal{E}_{\alpha}\right)\cup\mathcal{E}_{super}$ where $\mathcal{E}_{super}$ is the set of inter-super edges. The final numbers of nodes and edges are $N=|\mathcal{V}|=N_{0}+N_{super}$ and $E=|\mathcal{E}|\geq E_{0}$. Notice that this construction is equivalent to assume that the network has $L+1$ layers, where the extra layer is made of inter-super edges $\mathcal{E}_{super}$ and all nodes incident to them (without loss of generality, we assume that all the inter-super edges are treated equally). We denote it as super layer and this corresponds to $\alpha=L+1$, so that $\mathcal{E}_{L+1}\equiv\mathcal{E}_{super}$. We show an example of this structure in Figure 1.

Finally, we consider a coupling between layers as in morris2012transport that controls how the layers are linked. Specifically, we multiply the lengths of each edge by a factor $w_{\alpha}\in[0,1]$ that depends on what layer the edge belongs to. For convenience, we introduce $q_{e}\equiv q_{e}(\alpha)$ taking values $q_{e}=\alpha$, for each $e\in\mathcal{E}_{\alpha}$ and with $\alpha=1,\dots,L+1$. Using this, we define the resulting length as $\ell_{e}:=w_{q_{e}}l_{e}$. This ensures that edges in different layers can be navigated differently. If we interpret $w_{\alpha}$ as the inverse of a velocity, then $\ell_{e}$ is proportional to the time needed to travel along edge $e$, which can be seen as an “effective” length. When $w_{\alpha}<1$ and $w_{\gamma}=1$, a passenger takes less time to travel along an edge of length $l_{e}$ in $\alpha$ than one in $\gamma$. Typically, $\ell_{e}$ are small for inter-super edges. Nevertheless, one can tune the cost to travel along them by tuning $w_{L+1}$.

We consider the formalism of optimal transport theory, and in particular recent works that maps the setting of solving a standard optimization problem into that of solving a dynamical system of equations lonardi2020optimal; bonifaci2012physarum; santambrogio2007optimal; facca2016towards; facca2019numerics; facca2020branch; baptista2020network; bonifaci2020. Specifically, we model two main quantities defined on network edges: i) fluxes $F_{e}$ of passengers traveling through an edge $e$; ii) conductivities $\mu_{e}$, these are quantities determining the flux passing through an edge $e$. Intuitively, the conductivity $\mu_{e}$ of an edge can be seen as proportional to the size of the edge $e$. To keep track of the different routes that passengers have, we consider a multi-commodity formalism as in lonardi2020optimal, i.e. we distinguish passengers based on their entry station $a\in\mathcal{S}$, where $\mathcal{S}\subseteq\mathcal{V}$ is the set of stations where passengers enter, we denote with $M=|\mathcal{S}|$ the number of passenger types. With this formalism, we have that the fluxes $F_{e}$ are $M$-dimensional vectors, where the entries $F_{e}^{a}$ denote an amount of passengers of type $a$ traveling on edge $e$. The important modeling choice is that the conductivities $\mu_{e}$ are shared between passengers, thus they are scalar numbers contributing to the cost for all passengers’ types traveling trough $e$. This formalism can be equally applied to both edge types, intra-layer and inter-super edges.

We assume that fluxes are determined by pressure potentials $p_{u}^{a}$ defined on nodes as:

We model the amount of passengers entering a station $a$ with a positive real number $g^{a}$. For notational convenience, we define a $N\times M$ dimensional matrix of entries $g_{u}^{a}$ such that $g_{u}^{a}:=0$ if $u\neq a$, and $g_{u}^{a}:=g^{a}$ if $u=a$. Similarly, we define with $h_{u}^{a}$ the amount of passengers of type $a$ exiting at node $u$. Here the only constraint is that $h_{u}^{a}=0$ if $u=a$, to avoid unrealistic situations were passengers entering in one station exit from the same station. Finally, we define the $N\times M$-dimensional source matrix with entries $S_{u}^{a}=g_{u}^{a}-h_{u}^{a}$, which indicates the amount of passengers of type $a$ entering or exiting a station. Notice that for each $a\in\mathcal{S}$ we have $\sum_{u}S_{u}^{a}=0$, meaning the system is isolated, i.e. all the passengers of a certain type who enter the network also exit.

With this in mind, we enforce mass conservation by imposing Kirchhoff’s law on nodes. To properly enforce this constraint we need to consider all the edges, both intra-layer and inter-layer edges. This can be compactly written by considering the multilayer network signed incidence matrix $B$ with entries $B_{ve}=1,-1,0$ if node $v\in\mathcal{V}$ is the start, end of edge $e\in\mathcal{E}$, or none of them, respectively. With this in mind, Kirchhoff’s law can be written as:

Finally, we assume that the conductivities follow the dynamics:

where $q_{e}$ encodes the type of edge, as defined in Section II.1. The parameter $0<\beta_{q_{e}}<2$ is important as it determines the type of optimal transport problem that we aim to solve, as we describe in more detail later. Interpreting the conductivities as quantities proportional to the size of an edge, this dynamics enforces a feedback mechanism such that the edge size increases if the flux trough that edge increases, it decreases otherwise. This feedback mechanism has been observed in biological networks like the one made by the slime mold Physarum polycephalum Tero439; bonifaci2012physarum, which adapts its body shape to optimally navigate the space searching for food.

The important property of this dynamics is that its stationary solutions minimize a multilayer transport cost function:

where $\Gamma(\beta_{\alpha})=2\left(2-\beta_{\alpha}\right)/\left(3-\beta_{\alpha}\right)$ for all $\alpha$ and the $2$-norm is calculated over the $M$ entries of each $F_{e}$. This means that solving the systems of Eqs. 1 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