Supply Chain Exposure Half-Lives and Portfolio Selection Stability

Abstract

Quantitative models often refresh prices, returns, volatility, and macro variables at high frequency while carrying forward a commercial relationship graph from an earlier date. That creates a specific measurement problem: the model may be current in market space and stale in network space. This paper estimates the rate at which a historical supply-chain snapshot loses its ability to reproduce the contemporaneous network. The empirical exercise uses five directed Altsets relationship histories spanning semiconductors, aerospace, and consumer distribution, with quarterly observations from September 2021 through June 2026. For snapshot ages of 3, 6, 12, 24, 36, and 48 months, the analysis compares binary edge overlap, weighted overlap in Relationship Size, Supplier Revenue %, Customer Cost %, a bilateral dependence weight, exposure-rank correlation, and top-two portfolio membership. A one-parameter exponential decay curve fitted through 36 months yields a binary edge half-life of 19.2 months, while Relationship Size decays with a half-life of 8.0 months and bilateral dependence with a half-life of 6.7 months. Customer-side bilateral dependence decays much faster than supplier-side dependence in this panel. The separation between structural persistence and economic persistence is large enough to alter ranks and portfolio membership several quarters before many links disappear.

Keywords: supply chain data; temporal networks; exposure drift; portfolio rank stability; quantitative finance; feature decay

Problem

A network variable enters a quantitative model twice: through topology and through economic weight. The topology determines which firms are connected, while the weight determines how much of a supplier's revenue, a customer's cost base, or an estimated dollar relationship is attached to that connection. If a model formed at date $t$ uses a graph observed at $t-h$, the relevant error is therefore not captured by asking whether an edge still exists. A link can persist while its economic importance changes enough to alter a cross-sectional rank, a concentration statistic, a propagation coefficient, or a portfolio screen. The research object in this paper is the decay function of network information as snapshot age $h$ increases. The primary hypothesis is that weighted economic similarity decays faster than binary edge similarity. A secondary hypothesis is that portfolio membership decays at least as fast as the underlying economic weights because ranking errors occur when relatively small weight changes cross a selection threshold. A directional hypothesis is also testable: customer-side edges should decay faster than supplier-side edges when customer mix and order volumes adjust on shorter horizons than supplier qualification and sourcing relationships.

The distinction is economically material for any model that multiplies a market signal by a network exposure. Suppose a researcher computes a supplier shock score, customer-demand exposure, relationship-conditioned covariance, or graph-weighted factor at date $t$. If the market variable is current but the exposure matrix is stale, the resulting signal contains an interaction between current information and an outdated loading. That interaction can change sign or membership even when the historical adjacency matrix still looks familiar. A binary persistence statistic may therefore overstate the usable life of the network for finance. The empirical question becomes measurable: for an $h$-month-old snapshot, what fraction of current weighted exposure is preserved, how strongly are current ranks reproduced, and how often does a top-$k$ network portfolio contain the same names? A practical half-life can then be defined as the age at which a fitted similarity function falls to one-half of its date-zero value.

Prior Work

Finance has long treated customer-supplier links as economically relevant state variables. Cohen and Frazzini document return predictability across firms connected through principal-customer relationships, arguing that information about economically linked firms is incorporated with delay. Production-network research provides a complementary mechanism. Barrot and Sauvagnat show that idiosyncratic supplier shocks propagate more strongly when inputs are specific, and Carvalho, Nirei, Saito, and Tahbaz-Salehi document upstream and downstream propagation after the 2011 Great East Japan Earthquake. These studies establish that connection direction and economic dependence can affect outcomes. They leave a narrower measurement problem open for a researcher who carries a historical graph forward: how quickly does the graph itself cease to represent the exposure state that would have been measured contemporaneously?

Temporal-network research gives the statistical reason to separate edge survival from weight survival. Holme and Saramaki emphasize that network timing changes the behavior of processes on graphs, so an aggregate or static representation can discard information contained in edge activation times. Mazzarisi, Barucca, Lillo, and Tantari model interbank links with separate mechanisms for persistence and changing node fitness, showing that time-varying topology cannot be reduced to a static tendency to reconnect. Di Gangi, Bormetti, and Lillo extend that logic to sparse weighted networks and explicitly allow the probability that a link exists to be independent of its expected weight. That distinction is especially close to the present problem because a commercial relationship can survive while its dollar scale or dependence share moves substantially. Franch, Nocciola, and Vouldis likewise show in a financial-contagion setting that temporal representations can classify transmission differently from static aggregation. The missing object for commercial relationship data is an empirically calibrated horizon at which stale topology, stale weights, and stale ranks begin to diverge.

A binary survival ratio is a natural baseline. Leon and Miguelez measure the fraction of interbank links surviving across consecutive network states and longer windows, providing a direct persistence statistic for directed financial networks. That statistic answers whether counterparties continue to interact. For an investment model, however, two networks can have identical binary edges and very different exposure matrices. A supplier can remain connected to the same customer while the relationship doubles in dollar value, while its share of supplier revenue falls because the supplier grows elsewhere, or while the customer's cost share rises because the customer's cost base contracts. Each path produces a different financial loading from the same surviving edge. The present design therefore treats binary survival as the control object and asks how much additional decay appears once the commercial weights themselves are preserved.

Measure

Let the directed commercial network at date $t$ be $G_t=(V,E_t,W_t)$, with each edge $e=(s,c)$ running from supplier $s$ to customer $c$. Let $A_{e,t}$ equal one when the edge is observed at $t$ and zero otherwise. Three Altsets weights are used. $D_{e,t}$ is Relationship Size in dollars, $R_{e,t}$ is Supplier Revenue %, and $C_{e,t}$ is Customer Cost %. The first quantity measures economic scale, the second measures dependence of the supplier on the customer, and the third measures dependence of the customer on the supplier. Because $R$ and $C$ use different denominators, they are retained separately throughout the analysis. A symmetric bilateral dependence score is added for ranking applications:

The geometric mean gives a high score only when the relationship is economically meaningful from both sides, while avoiding a direct multiplication by Relationship Size that would mechanically reuse the same commercial scale embedded in the percentage measures. The empirical conclusions are also checked with the minimum, harmonic mean, and arithmetic mean of $R$ and $C$.

For two snapshots separated by $h$ months, binary similarity is the Jaccard overlap

For any nonnegative edge weight $x\in\{D,R,C,B\}$, weighted similarity is

This quantity equals one when the stale and contemporaneous exposure vectors coincide and approaches zero as common economic weight disappears or is resized. Absent edges receive weight zero, so births and deaths enter the measure together with resizing. To isolate resizing among relationships present in both snapshots, the analysis also computes the median absolute log error

The multiplicative interpretation is $\exp(L_x)-1$. This second statistic identifies a source of decay that a binary overlap cannot capture.

Rank preservation is measured by the Spearman correlation between $B_t$ and $B_{t+h}$ across the five selected edges, assigning zero to absent edges. The downstream portfolio object is intentionally simple. At each date, the two counterparties attached to the highest bilateral dependence scores form $P_t^{(2)}$. Membership fidelity at horizon $h$ is the average Jaccard overlap between $P_t^{(2)}$ and $P_{t+h}^{(2)}$. The point is to measure selection instability rather than returns. If two candidate relationships have current score gap $\Delta_{ij,t}=B_{i,t}-B_{j,t}$, a stale ordering reverses whenever the relative drift error exceeds that gap. As the variance of weight drift grows with $h$, rank and top-$k$ membership can decay faster than aggregate weighted overlap, especially near a portfolio cutoff.

A single decay curve summarizes the operational age of each object:

where $H_x$ is the information half-life in months. The curve is fitted directly to all snapshot-pair similarities from $h=0$ through 36 months. The 48-month comparisons are retained as a long-horizon check rather than given the same influence as denser short-horizon observations. Empirical network similarity can rise at some later horizons because an edge can disappear and subsequently reappear, so the exponential is a summary of average information loss rather than a structural law of relationship survival. The one-month similarities reported below are curve-implied values from the same fit; the raw history is quarterly.

Panel

The calculation uses five directed Altsets relationships rather than the entire historical set: SK Hynix to Intel, Intel to LG Electronics, Hyundai Glovis to Boeing, Boeing to Korean Air, and PepsiCo to Seven & i Holdings. The first two form a semiconductor case, the middle two an aerospace case, and the final edge a consumer-distribution case. Each selected active observation contains Relationship Size, Supplier Revenue %, and Customer Cost %, which allows binary and weighted comparisons to be made on the same relationship set. The quarterly window runs from September 2021 through June 2026. Across that window, the Hyundai Glovis to Boeing edge is continuously observed, while the other histories contain episodes of disappearance and reappearance. That mixture is useful for the research question because a half-life based only on continuously surviving links would mechanically suppress the topology component of staleness.

The selected relations also span economic settings with different reasons for persistence. Intel describes a global supply chain with thousands of suppliers, including some sole-source or geographically concentrated inputs, and states that it maintains close relationships with key suppliers. Boeing describes dependence on a large supplier base and a smaller set of sole-source suppliers, while delivery schedules depend on supplier performance. PepsiCo uses direct-store-delivery, customer-warehouse, distributor, e-commerce, and retail channels, with route-to-market choices depending on customer needs, product characteristics, and local trade practices; its filing also describes variation in customer programs. These operating structures do not identify the half-life mechanically. They provide an economic basis for expecting the rate of network change to differ across the three cases rather than treating every commercial edge as an exchangeable observation.

Decay

The first empirical result is a persistent gap between binary and economic overlap. At three months, mean binary Jaccard similarity is 0.741, while Relationship Size overlap is 0.649, Supplier Revenue % overlap is 0.680, Customer Cost % overlap is 0.556, and bilateral dependence overlap is 0.610. At six months, binary overlap is 0.590 and bilateral overlap is 0.383. The twelve-month binary statistic rebounds to 0.711 because some edges reappear at the same annual distance even after shorter intermittent gaps, yet bilateral overlap remains only 0.432. By 24 months, binary overlap is 0.503 and bilateral overlap is 0.290; by 36 months they are 0.388 and 0.222. The 48-month observations continue the separation, with binary overlap at 0.450 and bilateral overlap at 0.136. The nonmonotonic binary path is itself informative: network states can recur, while the economic weights attached to those recurring edges can be very different from the earlier state.

Source: calculations from Altsets historical relationship snapshots. The 1-month row is implied by the fitted decay curve.

The fitted half-lives make the binary-weight distinction more interpretable. Binary edge overlap has an estimated half-life of 19.2 months. Relationship Size is 8.0 months, Supplier Revenue % is 12.1 months, and Customer Cost % is 4.7 months. Bilateral dependence has a 6.7-month half-life. The top-two membership statistic decays still faster, with a half-life of 5.8 months. These values imply that the identity of a relationship can remain recognizable for roughly one and a half years while economically relevant exposure may become half as similar within two to four quarters. The three Altsets metrics also decay at different rates because each responds to a different denominator. Relationship Size moves with the commercial flow itself; Supplier Revenue % also moves with the supplier's total revenue; Customer Cost % also moves with the customer's cost base. A stale graph therefore contains several distinct clocks even before market data are introduced.

Resizing among surviving edges accounts for a material share of the decay. Conditional on an edge being present at both dates, the median absolute multiplicative error at 12 months is 25.6% for Relationship Size, 24.7% for Supplier Revenue %, and 20.8% for Customer Cost %. At 24 months, the corresponding errors are 38.9%, 26.2%, and 22.2%. By 36 months, they are 36.3%, 36.0%, and 64.5%. The pattern rules out an interpretation in which network staleness is driven only by births and deaths. Even when the stale snapshot correctly identifies a continuing relationship, the exposure assigned to that edge can be far enough from the contemporaneous value to alter a concentration estimate or a graph-weighted signal.

Ranks

Ranking deterioration is faster than the visual persistence of the graph. The bilateral dependence rank correlation averages 0.488 at three months and 0.143 at six months. It rises to 0.301 at twelve months as several relationships recur in familiar positions, then turns negative at 24 and 36 months. With five edges, those long-horizon rank correlations are case-panel diagnostics rather than population parameters. Their financial interpretation is still direct: a historical network can preserve several familiar links and fail to preserve which of those links are currently the largest. A model that only checks adjacency would treat those snapshots as similar even when an exposure sort has largely changed.

The top-two portfolio experiment makes the threshold effect visible. Membership overlap is 0.526 after three months, 0.315 after six months, 0.500 after twelve months, 0.250 after 24 months, and 0.208 after 36 months. Its fitted 5.8-month half-life is shorter than the 6.7-month bilateral-weight half-life and much shorter than the 19.2-month binary half-life. This ordering is consistent with the rank-crossing condition above. Portfolio membership can change after a modest relative move between two neighboring scores, long before the entire weight vector has lost half its similarity. In practice, the decay rate relevant to a quant process depends on the operation performed on the graph. A model that only needs a broad connected-versus-unconnected partition can tolerate older snapshots than a model that selects a narrow top-$k$ basket from economically weighted edges.

Asymmetry

Direction produces the largest cross-sectional difference in the panel. For the two supplier-side histories, fitted binary overlap has a half-life of 54.5 months and bilateral dependence has a half-life of 17.1 months. For the three customer-side histories, the corresponding estimates are 8.6 months and 5.3 months. The supplier-side binary estimate extends beyond the main fitting window, so its exact level is less informative than the ordering. The economic interpretation is that the selected supplier relationships retain identity for long periods while their weights still resize, whereas the selected customer relationships turn over and resize on shorter horizons. This pattern is compatible with supplier qualification, production integration, and sourcing frictions creating stickier relationship identities on the input side, while customer mix and order allocation can change more rapidly on the output side. The hypothesis would be contradicted in a broader sample if customer-side and supplier-side weighted half-lives converged after controlling for sector and relationship scale.

The sector cases reinforce the same distinction without supporting a broad industry ranking. In the two Intel relationships, binary half-life is 21.2 months and bilateral half-life is 14.2 months. In the two Boeing relationships, binary half-life stretches to 44.3 months while bilateral half-life is 10.7 months. In the PepsiCo to Seven & i case, binary and bilateral half-lives are 7.5 and 5.4 months. The aerospace case is the clearest example of why edge survival can be misleading for quantitative exposure: relationship identity is exceptionally persistent in the fitted curve, while the economic weight attached to those identities decays roughly four times faster. The consumer case shows a different process in which both topology and weight age quickly. Those differences accord with the operating descriptions in the companies' filings, but the paper uses them as case evidence for heterogeneous clocks rather than as sector-wide estimates.

Tests

Two tests challenge the main interpretation. First, the half-life gap survives removal of every individual relationship. Across five leave-one-edge-out runs, binary half-life ranges from 10.4 to 25.6 months, while bilateral half-life ranges from 5.4 to 11.6 months. Removing the episodic PepsiCo to Seven & i relationship lengthens the binary half-life to 25.6 months and the bilateral half-life to 8.1 months, so the economic-weight result is not generated solely by that edge's long inactive interval. Removing Boeing to Korean Air, which has large changes in customer dependence, produces a 25.5-month binary half-life and an 11.6-month bilateral half-life. In every leave-one-out panel, economic similarity expires before structural similarity.

Second, the bilateral result is not tied to the geometric mean. Replacing $\sqrt{RC}$ with the arithmetic mean produces a 5.4-month half-life; using the minimum of $R$ and $C$ gives 8.4 months; using the harmonic mean gives 8.5 months. All remain well below the 19.2-month binary half-life from the baseline panel. The exact number therefore depends on how bilateral dependence is defined, while the structural result survives reasonable alternative weighting choices. A genuine failure of the paper's main hypothesis would require economic-weight half-lives to approach the binary half-life under these alternatives or after removal of high-turnover edges. That pattern is absent in this five-relationship exercise.

Use

The fitted decay parameter can be converted into an application-specific refresh horizon. If a model requires at least a fraction $q$ of its date-zero similarity, the maximum snapshot age implied by the exponential fit is

For $q=0.70$, the binary graph can age about 9.9 months before its fitted similarity reaches 0.70. The corresponding ages are 4.1 months for Relationship Size, 6.2 months for Supplier Revenue %, 2.4 months for Customer Cost %, 3.5 months for bilateral dependence, and 3.0 months for top-two membership. These are direct consequences of the measured decay in this panel rather than universal update schedules. A quarterly refresh can be adequate for one weight and already stale for another; an annual refresh can preserve many link identities while substantially changing the ranking and magnitude of exposures used by the model.

Historical supply chain data can therefore enter a quantitative model with network age treated as part of the state vector. A researcher interested in shock propagation, covariance conditioning, pair selection, or exposure sorting can attach age to the last observed network weight instead of treating that weight as a fixed characteristic of the firm. One implementation is an age-adjusted quantity

when the latest observed edge weight is from date $\tau<t$. This adjustment is a conservative shrinkage device rather than an estimator of the unobserved current relationship. A richer specification could use edge-specific transition probabilities and conditional weight dynamics, as in the temporal-network literature, but the half-life calculation already supplies the minimum quantity needed to decide how much information an aging commercial exposure retains.

Conclusion

The five-relationship Altsets panel produces a clear separation between structural and economic persistence. Binary edge overlap decays with an estimated 19.2-month half-life, compared with 8.0 months for Relationship Size, 12.1 months for Supplier Revenue %, 4.7 months for Customer Cost %, and 6.7 months for bilateral dependence. Rank and top-two membership degrade faster still, with portfolio membership reaching a fitted half-life of 5.8 months. Supplier-side relationships persist longer than customer-side relationships in this case panel, and the aerospace example shows the largest gap between durable relationship identity and faster-moving economic weight.

The quantitative consequence is specific. A stale commercial graph can remain recognizable while ceasing to reproduce the exposure magnitudes and rankings used by a current model. Binary link survival therefore provides an upper bound on usable network age rather than a sufficient measure of it. For the selected histories, the economically relevant half-life lies closer to one or two quarters for customer-cost and portfolio-selection objects, and closer to two to four quarters for dollar size, supplier revenue dependence, and bilateral dependence. A historical supply-chain snapshot should be treated as a decaying quantitative state, with the decay rate attached to the exact network quantity used by the model.

References

Altsets supply chain dataset.

Cohen, L., and A. Frazzini. 2008. "Economic Links and Predictable Returns." Journal of Finance 63(4), 1977-2011.

Barrot, J.-N., and J. Sauvagnat. 2016. "Input Specificity and the Propagation of Idiosyncratic Shocks in Production Networks." Quarterly Journal of Economics 131(3), 1543-1592.

Carvalho, V. M., M. Nirei, Y. U. Saito, and A. Tahbaz-Salehi. 2021. "Supply Chain Disruptions: Evidence from the Great East Japan Earthquake." Quarterly Journal of Economics 136(2), 1255-1321.

Holme, P., and J. Saramaki. 2012. "Temporal Networks." Physics Reports 519(3), 97-125.

Mazzarisi, P., P. Barucca, F. Lillo, and D. Tantari. 2020. "A Dynamic Network Model with Persistent Links and Node-Specific Latent Variables, with an Application to the Interbank Market." European Journal of Operational Research 281(1), 50-65.

Leon, C., and J. Miguelez. 2021. "Interbank Relationship Lending: A Network Perspective." Physica A 573, 125922.

Di Gangi, D. D., G. Bormetti, and F. Lillo. 2022. "Score-Driven Generalized Fitness Model for Sparse and Weighted Temporal Networks." Information Sciences 612, 1226-1245.

Franch, F., L. Nocciola, and A. Vouldis. 2024. "Temporal Networks and Financial Contagion." Journal of Financial Stability 71, 101224.