Production process networks: a trophic analysis

In this section, we give a few definitions selected in the theory by Johnson and colleagues in [25, 27]. Let A be the adjacency of a graph G with the convention $a_{ij} = 1$ if there is an edge ² from node j to node i, and $k_i^{\mathrm{in}} = \sum_j a_{ij}$ and $k_i^{\mathrm{out}} = \sum_j a_{ji}$ the in and out-degrees of node i.

A walk is a ‘sequence of nodes such that every consecutive pair of nodes in the sequence is connected by an edge’ [31]. A directed walk respects the direction specified by the directed network. A path is a self-avoiding walk, that does not intersect itself.

The branching factor, whose role was evidenced in [18], is:

$$\begin{align} \alpha=\frac{\langle k^{\mathrm{in}}k^{\mathrm{out}}\rangle}{\langle k\rangle}. \end{align} \tag{ 1 }$$

In ecology [21], the notion of trophic level s_i of a node i was introduced to reveal the hierarchical ordering of species in food webs, as illustrated in figure 1:

$$\begin{align} s_i=1+\frac{1}{k_i^{\mathrm{in}}}\sum_j a_{ij} s_j. \end{align} \tag{ 2 }$$

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s_i is similar to the PageRank measure [32] defined for monopartite directed networks, with a normalization of the sum term by $k_i^{\mathrm{in}}$ instead of $k_j^{\mathrm{out}}$. Thus s_i is also akin to PageRank-related measures such as countries’ fitness and products’ complexity, as discussed in appendix E.

Equation (2) defines a set of linear equations that has a solution when each node belongs to a walk starting at a ‘basal’ node, i.e. a node such that $k_i^{\mathrm{in}} = 0$. Differences between trophic levels $x_{ij} = s_{i}-s_{j}$ and their standard deviation q, named ‘trophic coherence’, were introduced in [25]:

$$\begin{align} q = \textrm{std}(x_{ij}). \end{align} \tag{ 3 }$$

In [27] the relationship between q and the propensity for a network to include loops was examined. The authors introduce the ‘coherence ensemble’, which is the set of random graphs with a specified number of nodes, degree sequence, and coherence q. They compute n_ν the total number of walks of length ν in G, m_ν the total number of cycles of length ν, and c_ν the expected proportion of walks of length ν that are cycles, and their average values $\overline{n}_{\nu},\overline{m}_{\nu}, \overline{c}_{\nu}$ in the ‘coherence ensemble’. The authors show that the network ensemble belongs either to a ‘loopful’ (resp. ‘loopless’) regime depending on a parameter τ being positive (resp. negative), with the following definition:

$$\begin{align} \tau=\ln \alpha + \frac{1}{2\tilde{q}^2}-\frac{1}{2q^2}, \end{align} \tag{ 4 }$$

where α is the branching factor defined above, and $\tilde{q}$ is the average value of coherence in the ‘basal ensemble’ associated to G. This ‘basal ensemble’ is a restriction of the directed configuration ensemble with the additional constraint that the proportion of in-neighbors connected to non-basal nodes is kept fixed with value $k^{\mathrm{in}}_i L_\textrm B/L$, where L is the number of edges and $L_\textrm B$ is the number of edges connected to basal nodes.

Notably, in the large ν limit, the eigenvalue with leading real part noted λ₁ can be related to τ:

$$\begin{align} \overline{\lambda}_1 &= e^\tau \end{align} \tag{ 5 }$$

with $\overline{x}$ the average value of x in the ‘coherence ensemble’.

In this section we discuss basic definitions and problems in the field of LCI. Performing the LCI of a unit process consists in ‘the compilation and quantification of (its) inputs and outputs’ [33]. Inputs and outputs are usually called flows and each unit process is associated to a reference flow, i.e. its main output. A product system is a collection of such processes and flows.

For example, production of electricity can be modeled as a unit process: in a simplified way, to produce 10 kWh of electricity it uses 2 litres of fuel, and outputs 1 kg of carbon dioxide CO₂ and 0.1 kg of sulphur dioxide SO₂ [34].

There are several ways to represent unit processes and product systems [15]. Firstly, a unit process can be represented as a process vector, for example $\mathbf{p}_1 = (-2, 10, 1, 0.1)^\textrm T$ in the case of electricity production. The first dimension is associated to litres of fuel, the second one to kWh of electricity, etc$\ldots$. Negative values stand for inputs, and positive ones for outputs. A product system will then correspond to a set of m-dimensional vectors with m the number of flows, assembled in a rectangular matrix $\mathbf{P} = [\mathbf{p}_1,\ldots,\mathbf{p}_n]$, with n the number of processes. Secondly, an equivalent graph representation of a unit process is given in figure 2. The graph is directed and bipartite, since processes are only connected to flows, and conversely. It is weighted by the amount of inputs an outputs necessary to produce a given quantity of reference output, but each weight has a specific unit (for examples liters, kg, kWh, etc$\ldots$).

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A few more definitions are needed: elementary flows go from a process to the environment or reversely, intermediate flows (like products and wastes) are generated by processes. Processes can take both elementary and intermediate flows as input and output.

With the matrix representation, scaling up all inputs and outputs of process 1 by factor s₁, of process 2 by factor s₂, etc$\ldots$ amounts to multiplying P by a scale vector $\mathbf{s} = [s_1,\ldots, s_n]^\textrm T$. This representation is relevant to address the inventory problem, i.e. finding the scale vector s such that a demand flow f is met, with $\mathbf{f} \in \mathbb{R}^m$. For example, if 100 kWh of electricity are necessary, then p₁ should be scaled by $s_1 = 10$. Then this unit process will output 10 kg of CO₂ and 1 kg of SO₂ to the environment. Similarly if a product system P is scaled by vector s for some reason (for example meeting the demand in electricity, heat, and iron) then the total amount of environmental input and output flows can be readily computed. This constitutes LCA, that is the assessment of the environmental impacts of a product system meeting a certain demand.

Performing LCI and LCA, requires adequate datasets, prepared by experts and made available to process engineers. To put it simply, such datasets are large P matrices, with additional domain-specific metadata.

Then, we present briefly the common characteristics of LCI datasets and preprocessing steps required to build bipartite and monopartite graphs from them. Databases are selected because they are free to download, and have the following description taken from nexus.openlca.org:

• Agribalyse: ‘the French LCI database for the agriculture and food sector ($\ldots$) comprises LCIs for 2500 agricultural and food products produced and/or consumed in France’.
• ELCD: ‘(European reference Life Cycle Database) comprises LCI data from EU-level business associations and other sources for key materials, energy carriers, transport, and waste management’.
• Worldsteel: ‘This study contains global and regional LCI data for 16 steel products, from hot rolled coil to plate, rebar, sections, and coated steels’.
• Bioenergiedat: ‘Processes for bioenergy supply chains, with German background’.
• Ozlci: ‘The database inventory groups cover 958 (Australasian regional) supply chains’. It includes building products, chemicals, electric products, fabrics, farm and forest products, metal, minerals, as well as ‘utilities comprising use of freight, fuel, water and power by energy source and state grid’.

Some databases have a wide span, for example Agribalyse, which starts from the farm and goes to the distribution. Others have a narrower scope, for example Worldsteel.

In several databases, there is a convention that processes have a single product output, plus elementary flows. As in [15], we filter elementary flows since they do not connect vertices, which can result in processes having out-degree equal to one.

We do not perform monopartite projection because it loses information and can modify dramatically the networks’ properties [35]. In the case where process out-degree is equal to one, some properties (e.g. in/out degree) are conserved, but this is not the general case for other properties.

Additional preprocessing steps may be required, for example when modeling conventions result in spurious loops. This is documented for ELCD database in the case of transport ³ , and will be dealt with below by selective node removal.

Because of bipartivity, we cannot directly compare our results to those in [27]. For comparison purposes, a list of repositories of empirical networks was explored (see detailed list in appendix D). Among all examined empirical networks tagged as directed and bipartite (two-mode), most have a trivial trophic structure. This is mostly because in those cases all edges from mode A to mode B have the same direction (for example in a plant/pollinator foodweb, all edges go from plants to pollinators). Those networks were thus excluded from our comparison basis. On the contrary, metabolic networks from the BiGG database [37] that represent chemical reactions on one hand and metabolites on the other, are bipartite and directed and do not have a trivial trophic structure. Instead of considering the whole set of reactions and metabolites which would not correspond to a physiological phenomenon, we follow the standard practice in flux balance analysis which consists in maximizing an objective function (for example biomass production), which can be provided with the BiGG dataset. Optimizing the objective function leads to turning off some reactions, and getting rid of metabolites not involved at the selected operating point. After this preprocessing step, a bipartite network can be built from the remaining reactions and metabolites. Further preprocessing is added optionally, for comparison: thermodynamically infeasible cycles are removed using CycleFreeFlux [38], whose authors define cycles as ‘sets of reactions that together carry a flux that does not influence on the exchange reactions of the model ($\ldots$). These are metabolic “perpetual motion machines” and do not occur in biological reality’.

Lastly we mention the topic of finite size random graph samplers, that usually serve as a point of comparison in network science. Two of them are of interest in our case. Firstly, sampling from the basal ensemble in section 2.1 can be realized approximately using an off-the-shelf edge-rewiring algorithm, using a specific network g₀ as a seed (for example an empirically observed network, or a random graph with a degree sequence sampled from a specific distribution, such as the power law). Indeed, ensemble equivalence was exemplified in the monopartite case by [27, SI appendix §1.3] with finite size Erdös–Rényi and scale-free random graphs. When network size increases, this leads to $\langle q\rangle_{\textrm{samp}} = \tilde{q}$, where $\langle .\rangle_{\textrm{samp}}$ is the average under random sampling. Thus $\tau = \ln{\alpha}$, from equation (4). Hence τ in the initial graph g₀ is not preserved in general by such a random rewiring algorithm. The same can be observed in the bipartite case, with a rewiring algorithm that preserves both direction an bipartiteness, such as graph-tool [39]. Secondly, a sampler was proposed in [26] that allows to generate a random graph with a fixed value for q. But this sampler does not take a seed graph as an input, nor any degree sequence. Thus is does not allow to generate a random graph with specified q and degree sequence, which would allow to control α, and τ. Further, it has not been extended to the bipartite case. To conclude, random graph generators with a controllable degree sequence and trophic parameters at the same time do no exist in the literature so far, up to our knowledge, and this constitutes an interesting research direction.

In this section, we express m_ν and n_ν in the bipartite case, more specifically their average value $\hat{m}_\nu$ and $\hat{n}_\nu$ in the directed configuration model. Following the convention in [27], $\hat{m}_\nu$ does not concern simple cycles but includes multiple counts with different starting points in loops. Only the results are shown, and the full derivation is in appendix A.

Using the properties of adjacency matrices of bipartite networks, and the bipartite version of a directed configuration model, with an expected number of directed edges $p_{ij} = \frac{k_i^{\mathrm{in}} k_j^{\mathrm{out}}}{L}$ from node j to node i, we put A in the form:

$$\begin{align} A = \left( \begin{array}{cc} 0 & \mathbf{y\,v}^T \\ \mathbf{ux}^T & 0 \\ \end{array} \right) \end{align} \tag{ 6 }$$

and derive the average number of walks of length 2ν:

$$\begin{align} \hat{n}_{2\nu} = (\alpha_{xy} \alpha_{uv})^\nu \Bigg( \frac{L_{xy}}{\alpha_{xy}} +\frac{L_{uv}}{\alpha_{uv}} \Bigg) \end{align} \tag{ 7 }$$

with $\mathbf{x}^T \mathbf{y} = \alpha_{xy}$, $\mathbf{v}^T \mathbf{u} = \alpha_{uv}$, $L_{xy} = \sum_{ij} y_i x_j$ and $L_{uv}\sum_{ij} u_i v_j$. It can be compared to the expression of the monopartite directed configuration model in [27]:

$$\begin{align} \hat{n}_{\nu}^{\mathrm{mono}} = L \alpha^{\nu-1}. \end{align} \tag{ 8 }$$

Then the average number of cycles of length 2ν is:

$$\begin{align} \hat{m}_{2\nu} = 2 (\alpha_{xy} \alpha_{uv})^\nu \end{align} \tag{ 9 }$$

which is the counterpart of the following expression in the monopartite case:

$$\begin{align} \hat{m}_{\nu}^{\mathrm{mono}} = \alpha^{\nu}. \end{align} \tag{ 10 }$$

Interestingly $\hat{m}_{2\nu}$ is even: this is because it does not count the number of simple cycles but includes the same loops at different starting points, and each loop contains an even number of vertices.

Following [27] we finally define:

$$\begin{align} \hat{c}_{2\nu} = \frac{\hat{m}_{2\nu}}{\hat{n}_{2\nu}} = \frac{2}{\frac{L_{xy}}{\alpha_{xy}} + \frac{L_{uv}}{\alpha_{uv}} }. \end{align} \tag{ 11 }$$

Adapting equation (4) in the bipartite case requires an expression for $\tilde{q}$, the coherence in the basal ensemble. In short, we show that the expression found in [27] $\tilde{q} = \sqrt{\frac{L}{L_\mathrm{B}}-1}$ is only slightly modified, and the leading term is still $\sqrt{\frac{L}{L_\mathrm{B}}}$ in the common situation in which L is much larger than $L_\mathrm{B}$.

To see that, we keep the setting with N nodes, L edges and $L_\textrm B$ basal edges, and add bipartiteness. Left nodes include $N^L_\textrm b$ basal and $N^L_{\mathrm{nb}}$ non-basal nodes. Right nodes are exclusively non-basal and can connect only to left nodes, either basal or non-basal, as represented in figure 3. Basal edges go only from left basal nodes to right non-basal nodes. Non-basal edges are established between left non-basal and right nodes, in both directions. For consistency, left non-basal nodes must have in-degree greater than 1, which writes:

$$\begin{align} \tilde{k}^{\textrm{in},L}_{\mathrm{nb}} = \frac{L-L_\mathrm{B}}{2 N^L_{\mathrm{nb}}} \geqslant 1. \end{align} \tag{ 12 }$$

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In the basal ensemble, the proportion of in-neighbors connected to right nodes is kept fixed with value $k^{\mathrm{in}}_i L_\textrm B/L$. Right nodes receive a total of $\frac{L-L_\mathrm{B}}{2}+L_\textrm B = \frac{L+L_\mathrm{B}}{2}$ edges. Noting $\tilde{s}^\mathrm{R}$ the average trophic level of right nodes, and $\tilde{s}^\textrm L_{b}$ (resp. $\tilde{s}^\textrm L_{nb}$) the average trophic level of left basal (resp. non-basal) nodes, we get from the definition in equation (2):

$$\begin{align} \tilde{s}^\textrm L_{b} &= 1 \end{align} \tag{ 13 }$$

$$\begin{align} \tilde{s}^{\textrm{R}} &= {{1 + \frac{2 L_\mathrm{B}}{L+L_\mathrm{B}} +\frac{L-L_\mathrm{B}}{L+L_\mathrm{B}} \tilde{s}^\textrm L_{\mathrm{nb}}}}. \end{align} \tag{ 14 }$$

Similarly, non-basal left nodes receive edges from right nodes only and we get:

$$\begin{align} \tilde{s}^\textrm L_{\mathrm{nb}} = 1 + \tilde{s}^{\textrm{R}}. \end{align} \tag{ 15 }$$

This yields:

$$\begin{align} \tilde{s}^{\textrm{R}} &= \frac{L}{L_\mathrm{B}} +1 \end{align} \tag{ 16 }$$

$$\begin{align} \tilde{s}^{\textrm{L}}_{\mathrm{nb}} &= \frac{L}{L_\mathrm{B}} +2. \end{align} \tag{ 17 }$$

We remark that $\tilde{s}^\mathrm{R}$ has the same value as $\tilde{s}_{\mathrm{nb}}$ in the monopartite case [27], which explains why $\tilde{q}_\mathrm{b}$ is to leading order close to $\sqrt{\frac{L}{L_\mathrm{B}}}$ when L is much larger than $L_\mathrm{B}$. This is treated in detail in appendix B, and compared to numerical simulations.

Lastly, we explored the possibility that the numbers of edges leaving non-basal left layer and leaving the right layer are unbalanced, to reflect what is observed empirically with the LCI dataset. We report that indeed this modulates measured values for q. However, we were not able to find a simple yet accurate enough model for the observed behavior. Therefore in first approximation, we keep the balanced model below.

From equation (9) we have an expression for $\hat{m}_{2\nu}$ the average number of cycles of length 2ν in the bipartite directed configuration model. In section 3.2 we noticed that $\tilde{q}$ the coherence in the bipartite ensemble had approximately the same expression as in the monopartite case.

Following [27], and taking only loops with even length into account, we derive an expression similar to that in the monopartite case equations (4) and (5) $\lambda_1 = \max_i\{Re(\lambda_i) \}$:

$$\begin{align} \overline{\lambda_1} &= e^{\tau } \end{align} \tag{ 18 }$$

$$\begin{align} \tau &= \log \sqrt{\alpha_{xy} \alpha_{uv} }+ \frac{1}{2\tilde{q}^2} - \frac{1}{2q^2}. \end{align} \tag{ 19 }$$

The full derivation can be found in appendix C. As noted in section 3.1, $\sqrt{\alpha_{xy} \alpha_{uv} }$ has values that can be directly related to the monopartite branching factor α in certain particular cases, but not in general. As remarked in section 3.2 the bipartite definition $\tilde{q}$ can be well approximated by the usual monopartite value, in the particular case of balanced left and right layers.

In this section, the datasets presented in sections 2.2 and 2.3 are analyzed using the tools depicted in sections 3.1–3.3, adapted to the bipartite setting.

Table 1 shows that LCI networks are coherent, with an average q much lower than for other considered datasets, and are more likely in the loopless regime than other datasets. This leaves room for fluctuation inside the LCI category. For example, as shown in table 2, the Ozlci dataset has a trivial structure, since it occupies only two layers, and hence q = 0. Figure 4 presents two LCI networks embedded in 2d space ⁴ , with differing behaviors: the ELCD network is very coherent (q = 0.21), and more coherent than the corresponding average randomized networks ($q/\tilde{q} = 0.03$). The Bioenergiedat network is less coherent with value q = 0.99, and $q/\tilde{q}$ ratio just above 1.

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Table 2. Bipartite network trophic characteristics, LCI datasets. ‘Cycle’ is the number of unique elementary directed cycles, computed with graph-tool [39].

	N	NB	$\langle k \rangle$	q	$\tilde{q}$	α	$q/\tilde{q}$	τ	cycle	λ1
Agribalyse	31 698	1630	6.9	1.28	8.15	2.46	0.16	0.60	117	1.52
ELCD	894	104	11.5	0.21	6.72	3.42	0.03	−9.91	62	3.00
ELCD filtered	883	99	11.4	0.14	6.80	3.29	0.02	−23.46	0	0.03
Worldsteel	63	5	12.7	0.53	1.32	0.99	0.40	−1.51	0	$8.8e^{-4}$
Bioenergiedat	457	112	5.1	0.99	0.92	1.42	1.08	0.43	0	0.22
Ozlci	1914	957	1.0	0.00				$-\infty$	0	$1.9e^{-1}$

From the LCI literature, we expect a large number of feedback loops in the examined databases. Citing [34]: ‘Feedback loops occur frequently in industrial systems. For instance, mining of coal needs electricity, while production of electricity needs coal’. However, the majority of datasets are acyclic. Only Agribalyse and ELCD contain a small number of cycles compared to the edge number. In Agribalyse their occurrence is mostly related to seeds, which are both an input and output in plant growing. In ELCD, loops are associated with a few nodes, and filtering them is enough to remove all cycles, as explained in section 2.2.

This low number of loops has not been reported so far, up to our knowledge. It may arise from modeling bias, restricted scope, or from dataset selection bias. Also, recurring loops challenge numerical solvers in LCA not relying on matrix inversion. In the complex network literature, arguments of improved dynamic stability [18, 19] and transport [18] have been put forward to explain the lack of loops. In [20] the authors hypothesize that ‘instead the absence of feedback loops is a byproduct of a more inherent feature of networks: the existence of a preferred directionality’. In comparison, non-LCI networks introduced in section 2.3, contain a large number of cycles, not reported in table 3 because of prohibitive computational cost.

Table 3. Bipartite network trophic characteristics, other datasets. The number of cycles is not reported for this dataset because it is very high, and too time-consuming to compute exhaustively. Asterisks identify CycleFreeFlux preprocessing as explained in section 2.3.

	N	NB	$\langle k \rangle$	q	$\tilde{q}$	α	$q/\tilde{q}$	τ	λ1
iJR904	657	13	4.72	11.65	10.87	5.97	1.07	1.79	5.87
iJR904*	733	15	4.63	9.06	10.59	6.43	0.86	1.86	6.49
iSB619	623	12	4.70	11.90	11.00	5.92	1.08	1.78	6.04
iSB619*	658	17	4.70	8.46	9.48	6.27	0.89	1.83	6.54
iAF692	744	10	4.54	16.11	12.96	6.18	1.24	1.82	5.56
iAF692*	781	16	4.56	7.77	10.51	6.46	0.74	1.86	6.03
iND750	718	8	4.45	8.16	13.28	5.23	0.61	1.65	5.95
iND750*	824	20	4.42	10.70	9.49	5.53	1.13	1.71	6.47
iYO844	700	12	4.78	11.82	11.77	6.30	1.00	1.84	6.30
iYO844*	741	17	4.77	9.02	10.15	6.60	0.89	1.89	6.56
iAB_RBC_283	168	12	2.95	3.75	4.43	2.32	0.85	0.83	3.10
iAB_RBC_283*	285	21	3.39	3.79	4.69	3.68	0.81	1.29	4.42
iIT341	637	14	4.57	14.79	10.15	5.87	1.46	1.77	5.52
iIT341*	643	15	4.61	14.19	9.89	5.98	1.43	1.79	5.66
iNJ661	845	7	5.09	12.97	17.49	7.47	0.74	2.01	7.27
iNJ661*	925	19	5.04	10.07	11.03	7.99	0.91	2.08	7.89

Also the consistency of formulas in section 3.3, adapted from [27], that relate τ and λ₁, can be discussed. In figure 5, the leading eigenvalue λ₁ is plotted as a function of τ, for both datasets (LCI and non-LCI). Circles representing non-LCI data are well fitted by the dashed curve. Crosses representing LCI datasets are close to the exponential, except for the outlier value at $(\tau$, $\lambda_1) = (-10,3)$ which represents the ELCD dataset, and seems inconsistent with the curve. After filtering as explained in section 2.2 it is mapped to $(\tau$, $\lambda_1) =$ $(-23,0.03)$, which gives a satisfactory fit. This case is reminiscent of a remark in [20]: ‘typically loops are not independent as they can share some nodes. In particular, hubs are statistically more likely than other nodes to take part in loops’. Indeed removing just a few nodes changed dramatically the behavior from a coherence point of view. However such cycle-removing is of course not relevant for acyclic datasets, while in Agribalyse cycles are scattered across the dataset rather than concentrated. This raises the question of the robustness of coherence measures and will be discussed in section 4.

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In section 2.2 the topic of monopartite projection was evoked as it is easier to use off-the-shelf tools than to extend a theory to the bipartite case. This is the approach taken for example in [15], that may be justified by the particular nature of LCI databases, as noted in section 2.2. Several numerical experiments were run to try to find patterns in the effect of monopartite projection. First, this imposes the constraint that basal nodes must all belong to the same layer, more specifically to the layer chosen for projection. This problem may be mitigated using the new definitions of coherence in [29]. In the easier case where $k_{\mathrm{out}} = 1$ for processes and projection is done on the flow layer, we were not able to evidence a predictable behavior for coherence quantities. Sometimes the projected network has only two trophic values, which results in $q_{\mathrm{mono}} = 0$, although q was nonzero in the bipartite network. The case where $k_{\mathrm{out}}$ can take any value for processes is even harder to deal with.