Physics-Informed Implicit Representations of Network Flows
Anonymous Author(s)
AfﬁliationAddress email
Abstract
Flow networks are ubiquitous in natural and engineered systems, and in order tounderstand and manage these networks, one must quantify the ﬂow of commoditiesacross their edges.
This paper considers the estimation problem of predictingunlabeled edge ﬂows from nodal supply and demand.
We propose an implicitneural network layer that incorporates two fundamental physical laws: conservationof mass, and the existence of a constitutive relationship between edge ﬂows andnodal states (e.g., Ohm’s law).
Computing the edge ﬂows from these two laws isa nonlinear inverse problem, which our layer solves efﬁciently with a specializedcontraction mapping.
Using implicit differentiation to compute the solution’sgradients, our model is able to learn the constitutive relationship within a semisupervised framework.
We demonstrate that our approach can accurately predictedge ﬂows in several experiments on AC power networks.
Introduction
Network ﬂows are a fundamental aspect of modern society, from trafﬁc and communication networksto power and water distribution systems.
Many critical infrastructures are well-modeled as graphs,with edges that transport vital commodities [1].
Beyond infrastructure, network ﬂows are also centralto models in epidemiology, ecology, medicine, and chemical networks.
Their dynamics have beenwell-studied in compartmental systems theory [2].
Given the prevalence of ﬂow networks in naturaland engineered systems, predicting ﬂows in these networks is an important learning task that mayfacilitate monitoring, control, optimization, and protection of these networks.
While domain-speciﬁc tools to predict network ﬂows have been around for a while, the machinelearning community has only recently taken an interest in general-purpose models for networkﬂows.
[3] predicts edge ﬂows from partial measurements by making a smoothing assumption, i.e., byminimizing nodal ﬂow divergence.
[4] improves on this approach by adding a trainable regularizer thatcan incorporate side information.
Both of these approaches are centered on a notion of approximateconservation, i.e., that the net inﬂow to each node should be near zero.
Since conservation of mass isa universal constraint on network ﬂows, imposing this conservation law is an important step towardembedding physics into the model.
But the conservation law alone is not enough to uniquely determine ﬂow, which is why both [3] and[4] rely on heuristic regularizers to select the “best” conservation-respecting ﬂow.
In fact, physicalnetworks are often governed by a pair of physical laws: the conservation law, and a constitutiverelationship , which speciﬁes the magnitude and direction of each edge ﬂow based on “effort” variablesat each incident node (e.g., pressure or voltage).
For example, in DC circuits, currents are conservedaccording to Kirchoff’s current law, and Ohm’s law is the constitutive relationship that relates currentﬂows to nodal potentials.
The conservation law and the constitutive relationship together deﬁne theunique edge ﬂows (and nodal efforts).
1.1 Contributions
This paper proposes a model for network ﬂows that embeds both the conservation law and existenceof a constitutive relationship.
Our model, which we call an Implicit Flow Network (IFN), predictseach edge ﬂow using a trainable nonlinear function of latent nodal variables.
These latent variablesare constrained to a manifold wherein the conservation law is satisﬁed.
In addition to introducing
IFN, we offer the following contributions: (i) a contraction algorithm that is able to both evaluatethe IFN layer and backpropagate gradients through it, (ii) an explicit upper bound on the number ofiterations required by this algorithm, (iii) a rigorous theoretical comparison between IFN and thestate-of-the-art ﬂow estimation methods in [3, 4], and (iv) numerical experiments from several ACpower networks that indicate IFN can signiﬁcantly outperform these baselines on the ﬂow estimationtask.
Additionally, because IFN requires a nonlinearity with a constrained slope, we provide (v) anovel “derivative-constrained perceptron”, which is essentially a trainable activation function withupper and lower bounds on its slope.
1.2 Related Work
Network Flow Estimation Flows on graphs are a classical topic in computer science [5], and ﬂowforecasting has long been studied in speciﬁc domains like trafﬁc [6], but interest in the ﬂow estimationtask from a machine learning perspective appears to be relatively recent.
Deep learning algorithmshave been used to predict trafﬁc ﬂows [7, 8], but [3] and [4] appear to be the ﬁrst papers to proposemethods for domain-agnostic ﬂow prediction, based on the notion of divergence minimization.
Implicit Neural Networks IFN belongs to a growing class of models called implicit neural networks, which do not explicitly state the output of the model; rather, they describe a desired relationshipbetween the model’s inputs and outputs.
In the prevailing implicit framework, the output is deﬁned asa ﬁxed point of a trainable perceptron.
This approach was introduced in [9] as a “deep equilibriumnetwork”.
Subsequent work has developed new frameworks for ensuring the existence of the ﬁxedpoint and computing it [10, 11, 12, 13, 14].
Other types of implicit neural networks include neural
ODEs [15] and layers that solve convex optimization problems [16].
Graph Neural Networks Graph neural networks (GNN) are a diverse family of models for networkrelated learning tasks that incorporate graph structure directly into the model.
GNNs can typically beclassiﬁed into three types, in increasing order of generality [17, §5.3]: convolutional models [18, 19],attentional models [20], and message-passing models [21, 22].
Recently, [23] proposed an implicitgraph convolutional network.
Analogously, IFN can be interpreted as an implicit message-passing
GNN, with ﬂows serving as messages and latent nodal variables acting as an embedding.
1.3 Preliminaries and Notation
Given a directed graph G =
( V , E ) , the signed incidence matrix B ∈ {− 1 , 0 , 1 } |V|×|E| is the matrixwith entries
B i,e =   1 , i is the head of e − 1 , i is the tail of e 0 , else , ∀ i ∈ V and e ∈ E
For an undirected graph, the signed incidence matrix is obtained by assigning an aribtrary orientationto each edge.
Given a vector x ∈ R n , we use the notation [ x ] to denote the diagonal matrix diag( x ) ∈ R n × n .
Where such notation would be unclear (e.g., may be confused with brackets to indicate order ofoperations), we fall back on the diag( · ) notation.
We write x ⊥ to refer to the vector space that isorthogonal to x , i.e., the space { x (cid:48) ∈ R n : x (cid:62) x (cid:48)
= 0 } .
Given a positive deﬁnite diagonal matrix D ∈ R n × n , we write || x || 2 ,D to represent the weighted 2-norm || D 12 x
|| 2 .
[Figure]
Implicit Flow Networks
Implicit ﬂow networks are a family of message-passing GNNs that are inspired by the physics ofnetwork systems.
In many physical networks, nodes “communicate” through the exchange of acommodity, like power, water, or force.
These exchanges of commodities can be represented asﬂows across edges in the graph.
These ﬂows tend to be conserved, particularly in the steady statesof the system, so that Kirchoff’s current law is satisﬁed at every node.
Furthermore, these ﬂows aredetermined by a constitutive equation that yields the magnitude and direction of an edge ﬂow basedon states of the incident nodes.
Implicit ﬂow networks replicate these two constraints.
Deﬁnition 2.1 (Implicit Flow Network) .
An implicit ﬂow network (IFN) is a module with thefollowing components:
ﬁxed parameters 0 < d min ≤ d max ,2.
trainable parameters θ ∈ R r for some r , and3.
a family of differentiable functions h θ : R → R such that d min ≤ h (cid:48) θ ( y ) ≤
d max for ally ∈ R and θ ∈ R r , which we call ﬂow functions .
The module requires each of the following inputs:
a weighted, connected, undirected graph G =
( V , E , a ) with edge weights a ∈ R |E| > 0 , and
a supply / demand vector u ∈ R |V| such that (cid:80) i ∈V u i = 0 .
The module outputs the unique vector f ∈ R |E| for which there exists x ∈ R |V| such that
[Equation]
where B ∈ {− 1 , 0 , 1 } |V|×|E| is the signed incidence matrix of G , and h θ is applied element-wise.
We use the notation FN h,θ ( G, u ) to represent the solution f given inputs G and u , ﬂow functions h ,and parameters θ .
The IFN is illustrated in Figure 1.
The IFN’s only trainable component is its ﬂow function, parameterized by θ .
In practice, we will only make calls to the inverse of the ﬂow function when evaluating and backpropagating through IFN layers, so it is convenient to learn the inverse ﬂow function directly.
As the diagram highlights, IFNs are layers that can be situated in more complex architectures, withother models upstream estimating the supply / demand vector, edge weights, or even the topology.
For example, in power systems, demand forecasting is a very well-studied problem [24, 25], and onecan solve the economic dispatch problem to forecast power generation at each node [26], collectivelyleading to an estimate of the supply / demand vector. Theorem 2.2 (Well-Posedness) . IFNs are well-posed, i.e., there always exists an f ∈ R m andx ∈ R n that solve (1) – (2) , and f is unique.
We do not prove this theorem yet, as the existence and uniqueness is a trivial consequence of thetheory to be developed later in this section.
2.1 Evaluating the Implicit Flow Network
Our approach to evaluating the implicit ﬂow network is adapted from [27].
Any undirected graph Ginduces a direct decomposition of the space of ﬂow vectors R |E| : given the signed incidence matrix B ∈ {− 1 , 0 , 1 } |V|×|E| , the cycle space ker( B ) and cutset space Img( B (cid:62) ) are orthogonal, and [Equation] =
[Equation] B [Equation] Img( B (cid:62) )
We refer the reader to [28, §9.4] for a primer on cycle and cutset spaces.
Accordingly, we decomposethe vector f
= FN h,θ ( G, u ) as f = f cyc + f cut , where f cyc ∈ ker( B ) and f cut
∈ Img( B (cid:62) ) .
Thecutset component is readily from (1), since Bf = Bf cut = u implies that f cut
= B † u .
Then wemust analyze (2) to solve for f cyc .
Deﬁne a cycle projection matrix P ∈ R m × m as the obliqueprojection onto ker( B ) parallel to Img([ a ] B (cid:62) ) :
[Equation]
Based on this projection, we deﬁne a map T : ker( B ) → ker( B ) for all f cyc ∈ ker( B ) by
[Equation]
One can show that f cyc is the unique ﬁxed point of T , and that T is a contraction mapping, leading toa simple algorithm to compute this ﬁxed point.
Theorem 2.3 (Properties of T ) .
Consider an implicit ﬂow network with parameters d min , d max , andθ , with ﬂow functions h θ .
Suppose that the inputs G =
( V , E , a ) and u ∈ 1 ⊥|V| are given, and let
B ∈ {− 1 , 0 , 1 } |V|×|E| be the signed incidence matrix of G .
The following are true:
T is a contraction mapping with respect to || · || 2 , [ a ] − 12 , with Lipschitz constant
[Equation]
the sequence of iterates f ( k +1) cyc
= T ( f ( k ) cyc ) starting from any initial condition f (0)cyc ∈ ker [Equation]
)converges to a unique ﬁxed point f cyc , and
the output of the implicit ﬂow network is given by
[Equation]
Since f cyc exists and is unique, Theorem 2.3 immediately proves the well-posedness of the implicitﬂow network (Theorem 2.2).
Better yet, Theorem 2.3 provides a simple algorithm for computing thissolution: pick any f (0)cyc ∈ ker( B ) , repeatedly apply the map T until approximate convergence, thenadd B † u .
Some care is required when implementing this map.
Since P is a dense matrix with |E| 2entries, it is undesirable to explicitly construct the cycle space projection matrix for large networks.
Instead, in order to project a vector v ∈ R |E| , we can use the fact that
[Equation]
so the projection is evaluated as Pv = v − [ a ] B (cid:62) w .
Using this method of projection to implement
T , the ﬁxed point iteration to compute FN h,θ ( G, u ) is stated in Algorithm 1.
Algorithm 1 Evaluating the implicit ﬂow network.
1: B ← signed incidence matrix of G 2: f cut ← argmin f cut ∈ R m {|| Bf cut − u || 2 } 3: f cyc ← 0 m 4: ∆ f cyc ← ∞ 1 m 5: while || ∆ f cyc || 2 , [ a ] − 12 > (cid:15) do 6: v ← d min [ a ] h − 1 θ (cid:0) [ a ] − 1 f cyc + [ a ] − 1 f cut (cid:1) 7: w ← argmin w ∈ R n (cid:8) || B
[ a ] B (cid:62) w − Bv || 2 (cid:9) 8: ∆ f cyc ← v − [ a ] B (cid:62) w 9: f cyc ← f cyc − ∆ f cyc 10: end while 11: f ← f cyc + f cut 12: return f
Theorem 2.4 (Implicit Flow Networks, Forward Pass) .
Consider an implicit ﬂow network withparameters d min , d max , and θ , with ﬂow functions h θ .
Suppose that the inputs G =
( V , E , a ) andu ∈ 1 ⊥|V| are given, and let B ∈ {− 1 , 0 , 1 } |V|×|E| be the signed incidence matrix of G .
The followingare true of Algorithm 1, with a tolerance of (cid:15) > 0 :1.
for each iteration k = 1 , 2 , . . .
of the loop, let f ( k ) cyc represent the new value of f cyc deﬁnedon line 9; and let f (0)cyc = 0 m .
Then
[Equation]
the algorithm converges with at most k ∗ iterations of the while loop, where
[Equation]
) and ρ = || [ a ] h − 1 θ ([ a ] − 1 B † u ) || 2 , [ a ] − 12 ; and3.
the algorithm returns f ∈ R |E| , where
[Equation]
If evaluating h − 1 θ is sufﬁciently simple, then the most expensive step in the iteration is solving theordinary least squares problem on line 7.
Using a general-purpose solver, the complexity of thisoperation is roughly O ( |V| 3 ) .
But B [ a ] B (cid:62) is a sparse Laplacian matrix, so we can use a specialized
Laplacian solver that reduces the complexity to O ( |E| log k |E| ) for some constant k [29].
2.2 Computing the Gradients
In order to train the ﬂow function and any upstream models, it is necessary to backpropagate gradientsthrough the IFN layer.
We can perform this backward pass using implicit differentiation, and it turnsout that the gradients of FN h,θ ( G, u ) with respect to the parameters θ , a , and u can also be computedusing Algorithm 1, i.e., by writing the gradient as the output of an auxiliary implicit ﬂow network.
Theorem 2.5 (Gradients) .
Consider an implicit ﬂow network with parameters d min , d max , and θ ,with ﬂow functions h θ .
Suppose that the inputs G =
( V , E , a ) and u ∈ 1 ⊥|V| are given, and let
B ∈ {− 1 , 0 , 1 } |V|×|E| be the signed incidence matrix of G .
Let f = FN
h,θ ( G, u ) , and let w be ascalar entry of θ , a , or u .
We can compute the derivatives dfdw as follows.
Deﬁne a vector of ﬂow functions g : R |E| → R |E| by
[Equation]
Flow
Nodal Variable h
( y ) = DC Current Voltage y DC Power
Voltage y
AC Power (lossless)
Voltage Angle sin( y )
Water Flow Rate
Hydraulic Head sign( y ) | y | 0 .
Mechanical Force Networks
Position y where D ∈ R |E|×|E| is the diagonal matrix with entries
[Equation]
(9) and v = [ a ] h − 1 θ ([ a ] − 1 f cyc + [ a ] − 1 B † u ) .
Then
[Equation]
(10) (We use the notation · in place of θ , since g has no trainable parameters.)
Furthermore, the derivativeconstraint parameters d min , d max from the original implicit ﬂow network are valid for the new implicitﬂow network.
In other words, to compute the gradient with respect to a parameter, we perform a single evaluationof the implicit ﬂow network.
In order to compute the derivatives with respect to some parameter or input w , we ﬁrst evaluate thepartial derivatives ∂v∂w and the total derivatives dudw .
Then we construct the ﬂow functions g accordingto (8), and solve an implicit ﬂow network to ﬁnd dfdw according to (10).
It is trivial to evaluate dudw ,but for convenience, we provide the values of ∂v∂w below:
[Equation]
Comparison with Other Models
3.1 Physical Models
The equations (1)–(2) that specify IFN’s output are inspired by the governing equations of severalphysical ﬂow networks.
The steady-states of many network systems are determined by a conservationlaw and a constitutive relationship.
DC power networks are a familiar example: edge currents f areuniquely determined by Kirchoff’s current law Bf = u and Ohm’s law f = [ r ] − 1 B (cid:62) x , where uare nodal current injections,
r are resistances, and x are nodal voltages.
In lossless AC networks,active power ﬂows f satisfy the conservation of energy Bf = u and the power ﬂow equationf =
[ a ] sin( B (cid:62) x ) , where the edge weights a are functions of voltage magnitudes, line reactances,and tap ratios; and we re-interpret x as a vector of voltage angles [30, §6.4].
Equations of the form(1)–(2) show up in other models as well, including the Hazen-Williams formula in water distributionnetworks [31, Sec. 8.15], and discretizations of models in continuum mechanics [32, §16.9].
Table 1lists several of these networks, the physical interpretation of the nodal variables x , and the form ofthe ﬂow function h .
3.2 Bilevel Optimization
Both of the state-of-the-art methods for ﬂow estimation, from [3] and [4], use an optimization problemto predict ﬂows.
After a suitable transformation to incorporate external ﬂow injections u , we can state this optimization problem as
[Equation]
where λ > 0 , and q > 0 m is some vector of edge weights.
In [3], q = 1 m , while [4] allows q to bethe output of a neural network.
IFN is not explicitly an optimization problem, but it can be cast asone that is similar (11):
Theorem 3.1 (Optimization Form of IFN) .
Consider an IFN with ﬂow function h θ .
Suppose that theinputs G =
( V , E , a ) and u ∈ 1 ⊥|V| are given, and let B ∈ {− 1 , 0 , 1 } |V|×|E| be the signed incidencematrix of G .
Then the IFN output can be stated as the solution of a convex optimization problem:
[Equation]
Theorem 3.1 can be interpreted as a nonlinear generalization of the Thomson principle from electricalcircuits theory [33].
Interestingly, the theorem sets up a direct comparison between IFN and themodels in [3] and [4].
If the ﬂow function h θ is the identity map, then (12) can be simpliﬁed as
[Equation]
Ignoring the constraints from labeled ﬂows, we can interpret (11) as using a penalty method toapproximate the output of an IFN with a linear ﬂow function.
Thus, we have three distinct differencesbetween IFN and the optimization-based approaches:
IFN allows for a nonlinear ﬂow function, while [3] and [4] implicitly assume a linearconstitutive relationship;
IFN imposes ﬂow conservation as a hard constraint rather than an approximate constraint(which is a limitation if u is uncertain); and
IFN does not incorporate ﬂow measurements directly; rather, the model exploits thesemeasurements during training to learn the proper ﬂow function (and train any upstreammodels for the IFN inputs), making it less sensitive to noise in the labeled ﬂows.
Models for Flow Functions
In order to implement an IFN, it is necessary to parameterize its inverse ﬂow function h − 1 θ .
Since theﬂow function is essentially a trainable activation function, i.e., a scalar nonlinearity, simple modelsare likely to be sufﬁcient.
The main difﬁculty with selecting a ﬂow function is that its slope must bebounded by d min ≤ h (cid:48) θ ( y ) ≤
d max for all y ∈ R .
This section proposes a simple scalar nonlinearitythat is guaranteed to respect arbitrary upper and lower bounds on its slope.
Deﬁnition 4.1 (Derivative-Constrained Perceptron) .
Let k ∈ Z > 0 be a hidden layer size, let a, b, c ∈
R k be freely trainable parameters (encoded within the parameter vector θ ), and let σ be a nonexpansive activation.
Let p, q ≥ 1 such that p − 1 + q − 1 = 1 , and let ¯ d min ≤ ¯ d max ∈ R .
Then thederivative-constrained perceptron N ( x, θ ) is the 3-layer neural network deﬁned by
[Equation]
Intuitively, (L1) re-scales c so that the perceptron in (L2) is guaranteed to be non-expansive in x . Then(L3) re-centers and re-scales the derivatives of the perceptron from the range [ − 1 , 1] to [ ¯ d min , ¯ d max ] .
Theorem 4.2 (Derivative-Constrained Perceptron) .
Let N( x, θ ) be a derivative-constrained perceptron with ¯ d min ≤ ¯ d max ∈ R .
Then for all parameter values θ ,
[Equation]
[Equation]
Note that the values ¯ d min , ¯ d max in Deﬁnition 4.1 and Theorem 4.2 are distinct from the IFN parametersd min , d max .
Since we parameterize the inverse ﬂow function h − 1 θ in IFN, one shoudl set ¯ d min = d − 1maxand ¯ d max = d − 1min to implement h − 1 θ with a derivative-constrained perceptron.
Numerical Experiments5.1 Datasets
We selected 6 standard power network test cases.
The ﬁrst 4 test cases (IEEE-57, IEEE-118, IEEE-145,and IEEE-300) are synthetic transmission system test cases, while the remaining cases ACTIVSg200and ACTIVSg500 are similar to the Illinois and South Carolina power grids, respectively [34].
Eachtest case contains the topology and electrical parameters of the power network, as well as baselinedemands and power injections at each node.
While branch resistances are typically small, we setthem to zero to ensure lossless transmission lines.
We used the MATPOWER toolbox [35] to solvethe power ﬂow equations, then recorded the active power ﬂows on each branch ( f ), computed the netactive power injections at each node ( u ), and selected relevant electrical parameters as edge attributes(series reactance, tap ratio, and voltage magnitude at the two incident nodes).
See the supplementarymaterial for details.
5.2 Models and Experiment Details
IFN Architecture In order to use the IFN layer to predict power ﬂows, we created a two-layermodel.
The ﬁrst layer estimates positive edge weights a ∈ R |E| according toa e = exp ( L ( z e )) , ∀ e ∈ E where L is a linear module, and z e is the log-transformed vector of edge attributes.
The second layeris an IFN, with a derivative-constrained perceptron as the inverse ﬂow function ( k = 128 , d min = 25 ,d max = 2 , and p = q = 12 ), with a ReLU activation function.
The output of the IFN is used as thepredicted ﬂow vector.
Baselines We compared the IFN model against four baselines.
The minimum divergence method( Div ) from [3] minimizes the nodal divergence ||
Bf || 22 and a regularization term
λ ||
f ||
The bileveloptimization methods from [4] replace the uniform regularizer with a weighted regularizer
||
f || 22 , [ q ] ,where q is a vector of weights.
In Bil-MLP and Bil-GCN , q is the output of either a 2-layer MLPor GCN model with edge attributes as inputs (we use 64 nodes in each hidden layer with ReLUactivations).
Finally, in Bil-True , we specify q as the reciprocal of the coefﬁcient in the linearizedconstitutive relationship for AC power networks, so that Bil-True approximates (13) with a as theground-truth edge weight.
See the supplementary material for details and hyperparameter settings.
All of the baselines assume that nodal divergence Bf should be approximately zero, but nodes inpower networks inject and withdraw power according to the supply / demand vector u , resulting innonzero divergence.
Thus, when we evaluate the baselines, we transform the power network into adivergence-free network by introducing a “source node”, adding an edge from the source node to allnodes in V , and treating the entries of u as the ﬂows along each corresponding virtual edge.
Training We trained all models in a Google Colab notebook, using the Adam optimizer. We usedan initial learning rate of 0 . 01 , which we found to have good performance for all of the models. Training was terminated when the training loss had not decreased for 10 epochs.
We studied the transductive task of predicting unlabeled ﬂows, given that some labeled ﬂows in thesame network are known. If the edges E are partitioned into a labeled set E l and an unlabeled set E u ,the task is to predict the missing ﬂows { f e : e ∈ E u } given the labeled ﬂows { f e : e ∈ E l } . For eachof the 6 networks, we randomly selected a fraction of the edges to be labeled edges, and we trained
IFN on the labeled edges by minimizing the RMSE loss function [Equation] [Figure] IEEE
[Figure] IEEE-
[Figure]
Then we evaluated the same function on the unlabeled edges E u to compute the testing loss. Figure 2 reports the results for various fractions of labeled edges. IFN signiﬁcantly outperforms thebaselines on all of the networks when less than 80% of the edges are labeled. While the other baselinestend to improve as more labeled edges are made available for training, IFN achieves near-optimalperformance with as few as 10% of the edges labeled.
Conclusion
In this paper, we have introduced an implicit model for network ﬂows that incorporates physicsthrough a conservation law and through the existence of a constitutive relationship between ﬂowsand nodal variables.
We have demonstrated that a simple architecture using this model can learn toaccurately predict active power ﬂows in AC networks.
Future work may investigate more elaboratearchitectures using IFN as a layer, wherein the supply / demand vector, edge weights, or even thegraph topology itself could be predicted from upstream models, and the ﬂows themselves used fordownstream tasks.
Another interesting extension may be to apply our methodology to networks withhigher-order interactions, i.e., hypergraphs [36].
IFN also has some limitations that should be addressed in future work.
IFN assumes that the graphis undirected, which does not adequately model networks with unidirectional ﬂows (e.g., trafﬁc) orlossy ﬂows (e.g., resistive power grids).
IFN also assumes a constitutive relationship that dependson the difference between nodal variables.
This form appears frequently in physical systems, butin other network ﬂow models (like Daganzo trafﬁc models [37]), the constitutive relationship has amore general dependence on the nodal variables.
These limitations may be addressed with extensionsof IFN’s contraction algorithm.
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