https://orcid.org/0000-0001-6675-0355
Figures
Abstract
Detailed network models of social, biological, and other complex systems are often dense, which increases their computational complexity in simulations and analyses. To address this computational challenge, graph sparsification is used to remove edges while preserving desired network properties. For instance, distance backbones of weighted graphs are very sparse graphs that preserve community structure and spreading dynamics. Such backbones omit original graph edges that break a generalized triangle inequality for any given path-length measure and preserve all shortest paths of weighted graphs. They also best preserve node centrality ranks, as well as local and global dynamics in spreading phenomena. This superior performance of backbones is found by introducing the distance backbone synthesis (DBS), which progressively sparsifies weighted graphs according to a general family of nested distance backbones, whereby each edge is associated with the smallest distance backbone in which it appears. DBS provides a principled and natural method to span all ranks of sparsification possible while preserving connectivity, which allows for the precise study of (directed and undirected) weighted graph sparsification under multi-objective criteria. It provides an algebraically principled explanation of edge importance by revealing the precise topological space associated with each edge. The theory is demonstrated with a battery of social contact networks obtained from real-world social activity in different scenarios. This study also shows that the optimal preservation of node centrality and spreading dynamics happens for the distance backbone that obeys the generalized triangle inequality for the path-length measure , which removes more than half of the edges from the empirical networks studied.
Author summary
Detailed complex systems models, from social networks to biological interactions, are often represented as dense networks that are challenging to analyze. This can be overcome by using sparsification methods, which simplify dense networks to their essential components, like an X-ray view of a transportation network that keeps the essential routes for navigation. Our research systematically compares the ability of different methods to preserve structural and dynamical network features, and we introduce a novel approach to studying sparsification via distance backbone synthesis (DBS). This new approach provides a principled explanation of why certain connections are crucial, offering valuable insights for applications ranging from epidemic forecasting to understanding social influence. Most importantly, DBS identifies optimal network backbones, which not only speed up analysis and simulation but are also scalable and exact sparsification of networks. Such backbones can be achieved by removing more than half of all connections, those that have alternative shorter paths.
Citation: Pereira B, Costa FX, Rocha LM (2026) Distance backbones optimize spreading dynamics and centrality ranks in the sparsification of complex networks. PLOS Complex Syst 3(11): e0000137. https://doi.org/10.1371/journal.pcsy.0000137
Editor: Hocine Cherifi, Université de Bourgogne: Universite de Bourgogne, FRANCE
Received: March 2, 2026; Accepted: September 16, 2026; Published: November 3, 2026
This is an open access article, free of all copyright, and may be freely reproduced, distributed, transmitted, modified, built upon, or otherwise used by anyone for any lawful purpose. The work is made available under the Creative Commons CC0 public domain dedication.
Data Availability: The accompanying code is available at CASCI-lab/backbone-synthesis via https://doi.org/10.5281/zenodo.21309415. No data was generated by this study. The following existing data sources were used: SocioPatterns available via https://sociopatterns.org/datasets.html; and mobility networks from core-based statistical areas (CSBA) in the state of New York in the United States (US) available via https://github.com/CASCI-lab/SemiMetric_EpiSpread.
Funding: This work is financed by Portuguese national funds through the FCT - Foundation for Science and Technology, I.P., under the project 2022.09122.PTDC (https://doi.org/10.54499/2022.09122.PTDC). LMR was also partially funded by NIH National Library of Medicine Program grant 01LM011945-01 and the European Union’s Horizon Europe research and innovation programme under Grant Agreement No. 101186558 for the ‘Strategic Integration of Complex Networks and Systems for Advancing Biomedical Research’. The funders had no role in any stage of the study design to the manuscript preparation, nor decision to publish.
Competing interests: The authors have declared that no competing interests exist.
1. Introduction
Network analysis has been increasingly used to uncover complex patterns (e.g., central components, communities) in social, biological, and many other systems. Generally, a network has relational information (e.g., movement, contact) among multiple nodes (e.g., locations, individuals), all of which are represented in a graph with many edges (e.g., flux of passengers, frequency of contact) that connect node pairs. For example, in transportation systems, the flux of passengers among different locations can be mapped as a mobility network. It identifies key traffic routes and the most active areas, which can enable the development of effective traffic policies [1,2]. Likewise, a social system can be analyzed as a network of individuals connected by their frequency of contact, which can inform disease mitigation strategies [3,4]. More detailed system descriptions usually result in growing numbers of nodes and edges, which leads to challenges in computational resources for network analysis and higher burdens in theoretical explainability. Both call for the development of network sparsification methods capable of identifying subgraphs that preserve features of interest in the system’s network representation.
The complexity of determining which edges are essential and which are redundant for network inference is highlighted by the current multitude of methods available to tackle network sparsification. A commonly used initial approach to network sparsification is removing edges with small weights because they encode weaker interactions [5]. However, this method often leads to disconnected networks and fails to preserve the structural and dynamical properties of the original network because weaker edges may play key roles in these characteristics [6]. Another option, known as the disparity filter (DispFilt), compares the weights of edges against a null model to determine which edges are statistically significant [7]. It retains only those edges that do not deviate from the null model, which allows for a more informed reduction of the network, but it is biased by that chosen model. A less biased approach is the effective resistance method, which proceeds analogously to the computation of effective resistances in an electrical circuit [8,9]. It uses spectral graph theory to incorporate information about all parallel paths between pairs of nodes. A similar methodology that weights effective resistance a posteriori by the original edge weight has also been proposed [10]. Both of these methods, however, require the computation of the pseudo-inverse of the Laplacian matrix, which for large-graphs needs to be done using approximation algorithms. Other algorithms that use a mix of structural network properties and their statistical significance have been proposed for quantifying edge importance. These include the high-salience skeleton [11,12], omega-based link removal [13], and semi-metric distortion sparsification (SmD) [14]. All three rely on parameters used for thresholding, which tends to be network-specific.
Identifying the metric backbone, however, is an exact, algebraically principled and scalable network sparsification method. It removes edges that break the triangle inequality and are thereby redundant for shortest-path computation [15,16]. By generalizing the triangle inequality, even weighted directed networks can be sparsified to the union of all minimum spanning forests, which is the ultrametric backbone [17]. In the following, we explore other backbones [15] based on generalizations of the triangle inequality [18] and evaluate their preservation of desired network features.
In practice, representing a complex social or biological system as a network usually aims at inferring the importance of its constituent parts, the nodes and pathways, in information spreading (e.g., traffic, disease, news). In this context, it is important that network sparsification methods are capable of preserving the dynamics that happen in the original network [6,14]. For example, the effective resistance method preserves epidemic spreading under Susceptible-Infected (SI) [19], and Susceptible-Infected-Recovered [10] dynamics. Moreover, when shortest-path redundancy is accounted for via the metric backbone and SmD methods, sparser and more explainable subgraphs preserve SI [6] and Susceptible-Infected-Susceptible (SIS) [14] dynamics. That is, given that the length of a path is found by distance addition, the main pathways for information spreading in complex networks are those nearing the shortest ones.
Based on the idea of removing shortest-path redundancy, we introduce a novel method for progressive network sparsification, which we call the distance backbone synthesis or DBS, where edges are ranked by the distance backbone to which they belong. DBS is able to identify the main subgraph for dynamical processes because its edge ranking encodes the topological space in which these edges become a shortest path. We evaluate the capabilities of our method to preserve SI spreading dynamics and eigenvector centrality, the latter being an indirect marker of network response to many dynamical processes [20–22]. We compare our results with state-of-the-art network sparsification methods (DispFilt [7], wEffRes [10], and SmD [14]) and reveal that DBS explains the topological space on which dynamical processes occur via the optimal subgraph that preserves the dynamics. Moreover, we uncover that the metric and product backbones [15] are non-parametric strict and lenient network sparsification bounds, respectively.
2. Background
A weighted network representation of a complex system, G, is made up of a set of nodes, X, that are connected pairwise by edges, which are quantified according to a measure of either distance or proximity. Proximity edge weights quantify a notion of normalized interaction strength whereby the graph adjacency matrix has elements , with
denoting that two nodes
do not interact and
denoting that they have the strongest possible interaction. Distance edge weights quantify a notion of dissimilarity whereby smaller values correspond to stronger interactions and the adjacency matrix elements are
, with
denoting that
do not interact.
Importantly, is not strictly a distance, for it may not obey a triangular inequality and can be asymmetric (
). In fact, this lenient interpretation of distance is what allows us to find distance backbone subgraphs for directed [16] and undirected [15] networks. Both proximity adjacency matrix elements,
, and distance adjacency matrix elements,
, can represent the same network G via to a continuous, strictly decreasing map,
, that obeys the respective boundaries,
and
. This makes the two quantities isomorphic [18]. A canonical example is
which we use henceforth as the edge weights of G, unless explicitly noted otherwise.
Identifying the distance backbone subgraph of a complex network requires determining if there is an alternative shortest path for all existing edges [15,16]. Given any pair of nodes , a path is an ordered set of non-repeated nodes in which the initial element is
, the final element is
, and for any two consecutive nodes there exists an edge connecting them. The path-length
for a given path
can be given by a sum of the distance weights over the path’s edges, traditionally
This implies that the cost associated with going from to
is additive, where the “cost” is a surrogate intuitive term for mathematical distance. However, as was stated in [15,18], there are infinite other ways of computing a path length, or associated cost, that do not necessarily rely on the summation of the edge distances, for example, a multiplicative cost as shown in Fig 1. These infinite other ways were first presented in [18] and are called triangular distance norms (td-norms). Formally, a td-norm is a binary operation
that has a neutral element, g(a, 0) = a; is associative, g(a, g(b, c)) = g(g(a, b), c); is commutative, g(a, b) = g(b, a); and is monotonous,
for all
. Conceptually, these axioms mean that a path-length measure should i) not be altered by a distance length of 0 (neutral element), ii) not depend on the order in which the edges are traversed (associativity and commutativity), and iii) preserve the relative length order between paths passing via a similar edge set (monotonicity). With this in mind, we can rewrite the traditional path-length formula of Eq. 2 as
In the context of the triangle inequality, what matters then is to compare the direct with the smallest indirect cost. In this example, the cost of going from to
and then
is
according to the path-length measure in Equation 11. The cost can be metric (Eq. 2, red line), product (blue area), or many other values (see Sec. 2 for an overview). Moreover, if there exists a direct edge between
and
, and its dissimilarity measure is larger than the cost, shown on the right, it is considered semi-triangular.
Note that due to the associativity and commutativity of td-norms we can extend the notation to accept n-ary inputs instead of just binary, that is, . The traditional sum g(a,b) = a + b obeys all the axioms of a td-norm as well as other notable examples like the euclidean sum
, the distance product
and the maximum function
. Given that a td-norm defines how the length of a path is computed, the shortest-path length between a pair of nodes
is
The T superscript refers to the transitivity (not transpose) induced in the shortest-path computation because the all-pairs shortest path problem results in the transitive distance closure of the original graph [18]. This means that , which allows us to define a neighborhood for every
that is a topological space induced by the path length measure g [18,23]. This space is composed of the edges that satisfy
, for they obey the generalized triangle inequality imposed by the td-norm g,
These edges compose the distance backbone of a graph, , associated with the td-norm g [15,16]. Furthermore, how much each edge in G breaks the generalized triangular inequality of Eq. 5 can be quantified by the semi-triangular distortion,
The edges in the backbone satisfy
and are said to be triangular; the remaining edges satisfy
, and are said to be semi-triangular [15,16]. As such, triangular edges are those that have a direct cost lower than, or equal to, the cost of going along any alternative indirect path.
The particular distance backbone obtained using is the metric backbone, denoted by
. In this case, we say that the edges in the backbone are metric, instead of triangular, and those with semi-metric distortion
are semi-metric. The metric backbone has been shown to maintain community structure and preserve SI epidemic spreading [6]. Also, it has been demonstrated that the SmD identifies influential edges in spreading dynamics because the removal of edges with higher
has negligible effects on the transmission timescale and order of infection. In other words, using the SmD method, a complex network can be progressively sparsified in decreasing order of semi-metric distortion while preserving dynamical features [14].
The two limiting cases of distance backbones are the ultrametric backbone , found by setting the path length measure to the maximum td-norm, g = gmax (Eq. 7), and the drastic backbone Bdrastic, found by setting the path length measure to the drastic td-norm, g = gdrastic (Eq. 8). The former is the smallest possible backbone and comprises the union of all minimum spanning forests [17], while the latter is the unaltered original graph G [15],
These limiting cases arise because all td-norms, g, are bounded by gmax and gdrastic everywhere on [15,18]. In general, we have
Therefore, parameterizing g such that it spans the space of td-norms from gmax to gdrastic will give us a sequence of distance backbones from the smallest subgraph to the original network. In the following, we explore the consequences of such parametrization in inferring structural and dynamical network properties.
3. Results
3.1. Establishing the distance backbone synthesis
To span the different ways in which a path length can be measured in complex networks [18], we introduce the novel sparsification method, DBS. It is a systematic way of computing all distance backbones within a range of parameterized triangular distance norms, . Without loss of generality we can define a
such that i) for
we have
in its entire domain,
, and ii) it is bounded by gdrastic and gmax. Also, for an edge from
to
to be in the backbone
it must obey
, so it follows that
, which shows that the same edge is also in the backbone
. In fact, the backbones found via this ordering of
are nested, forming a chain
This newly found encapsulation of backbones, ranging from the smallest backbone possible to the whole network, implies that for each edge in the original graph, there is a first, smallest for which that edge becomes part of the corresponding backbone. Moreover, because
is generally a continuous parameter and there is a large heterogeneity of distance weights in real-world networks, each edge has a unique
. This induces an ordering of the edges that acts as a synthesis of all possible distance backbones from the family of path-length measure
. Also, because
enforces a topology via its shortest-path closure, finding
for each edge also provides all the shortest paths of the network in every topological space it covers. While we focus on particular cases here, Supplementary Sec. C in S1 Text provides additional
examples.
Consider the Dombi td-norm family inspired by fuzzy logic [18,25],
We have a continuous parameter such that i) when
the td-norm converges to
, whereas when
it converges to
; and ii)
for
in the domain
. An advantage of this parametrization, beyond its algebraic simplicity, is the fact that it generalizes the usual sum, as g1 (x,y) = x + y, which is the canonical operation used in the computation of shortest paths. It also generalizes pathfinder networks [26], which are often used in the analysis of high-dimensional numerical datasets in the context of machine learning and clustering algorithms [27,28]. This goes beyond the Minkowski r-metrics (or
-norms), with
, and allows any positive real value,
.
DBS allows us to span through different levels of sparsification by varying a threshold below which an edge with (where
is the value of
found using Equation 11) is in the sparsified subgraph (Fig 2). Note that the adjusted relative size of subgraph
, which ranges from
for the ultrametric backbone to
for the entire graph, is the complementary cumulative distribution function of
. Its sigmoidal shape on a semi-log scale suggests a log-normal distributed
. We find excellent agreement with the log-normal distribution when we perform the curve fit of
for the empirical social networks described in Sec. 5.1; the lowest R2 is 0.98, as shown in Supplementary Table B in S1 Text. The average mode of these log-normal distributions is
, meaning that most edges in the examined networks lie in some sublinear or fractal topological space (
). This is expected considering that their metric backbone is much less than 30% of the network. The average mode value also indicates that
suffices to considerably sparsify many social contact networks.
(Above) Important backbone subgraphs in this sparsification: drastic , which is the original graph G,
, metric
, euclidean
, and ultra-metric
. (Below) The relative size,
of the subgraph found via the backbone synthesis based on path-length measure in Equation 11 with varying
. When
there is no sparsification, and when
there is the largest possible sparsification.
While DBS creates a new parameter of edge importance based on distance backbones, other sparsification methods also quantify edge importance via different metrics. Most popularly, edge importance is determined in weight thresholding (W) by interaction strength and in DispFilt [7] by the statistical significance relative to a null model. For dynamically relevant edges, importance is assigned by wEffRes [10] and SmD [14]. A runtime comparison among those methods is shown in Supplementary Fig B in S1 Text. The results show that the current implementation of DBS is two orders of magnitude slower than identifying a single backbone, which runs at a time comparable to the other state-of-the-art methods. Nevertheless, no single sparsification takes more than 2 minutes.
Beyond runtime, it is also important to consider the ability of each sparsification method to not break the network into disconnected components. For instance, in the formulation of W, DispFilt, and wEffRes (see Methods 5.2) there is no guarantee of connectivity preservation, even though these methods have been found empirically to maintain it up to a certain point. On the other hand, methods based on the existence and preservation of shortest paths, like SmD and DBS, always preserve connectivity because they ensure at least one path between any two nodes. In the case of SmD, connectivity is guaranteed only down to the metric backbone size. At this point, all edges are indistinguishable under , and removing any edge under these conditions might disconnect the network. However, DBS preserves connectivity throughout the largest possible sparsification range, all the way down to the ultrametric backbone,
[17]. This connectedness is a requirement to preserve the shortest path length distribution. Supplementary Sec. B in S1 Text shows empirically that only SmD and DBS guarantee preserving connected metric and ultrametric shortest path lengths; the other sparsification methods only achieve this preservation in particular cases.
Below we further compare how these different sparsification methods preserve the structural and dynamical information of the original network.
3.2. Distance backbones preserve centrality and dynamics
One of the most widely used measures of node importance is eigenvector centrality because it accounts for the quality and quantity of the pathways in which a node is involved [29]. Therefore, one expects the absolute centrality value to be modified by network sparsification methods as edges are removed from nodes in the process. Nevertheless, in practice, the ranks of eigenvector centrality in a network are a more appropriate marker of node importance in various dynamical processes, due to their absolute variability [20–22]. Thus, we identify which sparser version of the original network best preserves the original centrality rank, , of the nodes. We also compute the preservation of Katz centrality [30] in Supplementary Sec. D.1 in S1 Text. This centrality metric introduces an attenuation parameter that allows for a transition from degree to eigenvector centrality, and has been shown to be relevant in various network analyses [10,31,32].
Node centrality is best preserved by sparsification methods that remove shortest-path redundancy. We summarize this finding for all 12 social contact networks analyzed by measuring the area under the curve, AUC of up to a maximum sparsification,
, as described in Sec. 5.3.3. Each empirical network corresponds to a dot in the boxplots of Fig 3 for each sparsification method. Overall, the distribution of
is statistically higher when edge importance is measured by DBS or SmD than with wEffRes, DispFilt, or W, as shown by the statistical tests in Fig 3 (Left). Because
is bounded from above by 1, removing shortest-path redundancy, either via SmD or DBS, is a more generally relevant methodology to sparsify complex networks while preserving eigenvector centrality rank. These findings are generalized for random social networks and mobility networks in Supplementary Sec. E in S1 Text. It also shows that DBS can find sparser connected subgraphs than any other method for the same desired
for random social networks (Supplementary Fig T in S1 Text). Similar findings are observed in terms of Katz centrality, where DBS performs comparatively better than wEffRes, DispFilt, and W, as measured by the Wilcoxon test. Together, these results show the robustness of sparsification methods that remove shortest-path redundancy while consistently preserving node centrality across diverse applications.
From the 12 social contact networks described in Sec. 5.1, we have a distribution of the area under the curve for different sparsification levels of (Left) eigenvector centrality rank, , (Middle) average time of infection,
, and (Right) time at which half of the nodes become infected in the original network relative to the same time measured in the sparsified network,
(details in Sec. 5.3). We compare the performance of DBS against each of the other sparsification methods discussed in the text (see legend) via the Mann-Whitney and Wilcoxon statistical tests and obtain the annotated
. The former tests whether the distribution of the performance metric is larger for DBS than another method, while the latter compares whether the difference between the performance metric for DBS and another method is positive.
Network sparsification based on shortest path redundancy optimizes the preservation of spreading dynamics. At a microscopic scale, we quantify the order in which each node gets infected, , in an epidemic spreading agent-based simulation of an SI model as described in Sec. 5.3.2. Fig 3 (Middle) summarizes our findings for all 12 social contact networks analyzed via
up to a maximum sparsification,
, as described in Sec 5.3.3. It is striking that a more informative method such as DispFilt performs just as well as the simple W, though the other methods have higher performance. DBS performs better than both wEffRes and SmD when the networks are individually compared, as shown by the statistically significant Wilcoxon test result. At a macroscopic scale, we quantify the timescale of the SI spreading,
for all 12 social contact networks via the distribution of the area under the curve
in Fig 3 (Right). The methods wEffRes, SmD, and DBS have more superior performances than DispFilt and W. The high performance of wEffRes is not surprising, as this method involves approximating the network Laplacian, which encodes a diffusion process in the network, so it is expected to capture the spreading timescale well. However, this comparative performance is mainly driven by infections starting in high centrality nodes. If the infection initiates in any node, SmD and DBS have consistently better comparative performance, as seen by the Wilcoxon test result in Supplementary Fig L in S1 Text. Moreover, the performance of wEffRes is significantly worse in random social networks (Supplementary Fig R in S1 Text). Ultimately, the shortest path redundancy that informs sparsification via SmD or DBS is a more traceable high-performing methodology because it does not change the edge weights in an irreversible operation as wEffRes does (See Sec. 5.2).
The superior performance of methods based on distance backbones, SmD and DBS, applies to a diverse set of empirical networks. In supplementary material, we show our findings are also supported for empirical networks measured from random social encounters and mobility data (See Supplementary Sec. E in S1 Text). For instance, both SmD and DBS performed overwhelmingly better than other methods when we considered networks with more random connectivity patterns (See Supplementary Sec. E.1 in S1 Text). In this scenario, W and DispFilt performed extremely poorly because they removed bridges very early on in the sparsification process, which disconnected the networks and hindered their spreading ability. The variability in the performance of wEffRes implies that it is less generalizable than SmD and DBS. Therefore, this demonstrated higher performance of sparsification methods that remove shortest-path redundancy highlights the wide applicability and benefits of distance backbones in distinct types of networks.
3.3. Distance backbone synthesis reveals optimal backbones
Having established DBS as one of the best network sparsification methods for preservation of network structure and dynamics, we investigate if there is an overall distance backbone capable of optimizing the preservation of such features. This task is achieved by optimizing for , because measuring the length of paths in a network using Equation 11 with
results in the backbone subgraph
. The optimal
distributed over the empirical network is shown in Fig 4. The optimization maximizes the weighted sum of the eigenvector centrality ranking,
, infection timing ranking,
, infection timescale,
, and sparsity,
, over various scenarios (see Methods Section 5.4). Our results show that an optimal backbone would be between the product backbone, found by the path length measure
, and the metric backbone, found by the path length measure g(x, y) = x + y. Despite neither being an overall optimum, they give the lenient and stringent bounds for a sparsification that preserves network features.
We measure the optimal per network, found via a weighted sum of the eigenvector centrality ranking,
, infection timing ranking,
, infection timescale,
, and sparsity,
, over various scenarios described in Section 5.4. The boxplots show the optimal
over the 12 social contact networks considered. The particular cases of the metric backbone (light green) and the product backbone (light blue) are highlighted. The latter is a small range of values because it can only be defined in another td-norm family, as discussed in Supplementary Sec. C in S1 Text. The median and interquartile range of the median
over the 12 networks are shown in black.
Our search for an optimal backbone has also revealed that most connections can be removed while still preserving multiple network features. As mentioned in Section 3.1, most of the edges in the social networks considered have . In addition, Fig 4 shows that
is a higher lower bound than that of the product backbone (
) for a balanced optimum (see Methods Section 5.4 for details). Therefore, setting
achieves large amounts of sparsification while still preserving at least two of the three performance metrics, eigenvector centrality,
, infection timing rank,
, or infection timescale,
. In addition, all three performance metrics are above 80% of their original value for
(Supplementary Fig P in S1 Text).
Each optimization scenario has a characteristic topological space identified by the DBS parameter . However, due to the computational costs in measuring
, we recommend that users of this methodology choose a priori the desired topological space depending on the sparsification leniency. (We provide specific
for given percent thresholds of the performance metrics in Supplementary Fig P in S1 Text.) A more computationally efficient alternative to DBS is SmD [14]. A multi-objective optimization (see Supplementary Fig O in S1 Text) finds that a lenient sparsification results from removing edges with SmD
, while edges with
are needed to satisfy multiple objectives. The main difference in the optimal subgraph for both DBS and SmD is in the spread of their distributions, where DBS provides a more precise threshold for an optimal subgraph across networks.
4. Discussion
In this study, we investigate how distance backbones can be used to create sparser yet dynamically and structurally faithful representations of complex networks. In particular, DBS identifies which topological space each edge belongs to based on how the length of paths is measured (Equation 11). The topological space is identified by the edges in the distance backbone for a given synthesis parameter, . Higher values of
induce more sparsification because they assign shorter distances to indirect paths. Different backbones, induced by path length measures such as Equation 11, are the main subgraph for different attributions of indirect path costs. In other words, the topological space defined by
constrains the cost of indirect paths of complex networks.
Network sparsification facilitates the analysis and simulation of social, biological, and many other complex systems [14]. However, for meaningful analysis, a reduced representation should maintain a system’s function. We evaluate the ability of state-of-the-art sparsification methods to preserve eigenvector centrality ranking and SI spreading (Section 3.2) as a proxy for maintaining a system’s function. In this task, DBS and SmD allow for a progressive network sparsification that outperforms more traditional methods [5,7,10] across many networks with diverse connectivity patterns. The similar performance between DBS and SmD is because they look at shortest-path redundancy in related ways [14,15]. DBS informs about the topological space to which each edge belongs, while SmD informs how distorted an edge is from the metric topological space. Therefore, DBS offers a deeper topological understanding and theoretical grounding of the structure of and dynamics on networks, while SmD offers practical, timely, and quality network sparsification.
Another noteworthy aspect of our analysis is the large diversity among the networks studied, which have very different sizes, connectivity, and density. Since we focus on epidemic spreading simulations, we have studied empirical social contact and mobility networks for their empirical relevance. On the other hand, those networks not only have different sizes, but also different modular organization ranging from random connectivity in an exhibit [33] to highly modular groups of individuals in classes [4,24,34–36]. Despite an incomplete understanding of how network connectivity, density, and modularity affect sparsification methods, our results from comparing the performance distributions show that removing shortest path redundancy is a simple way to sparsify networks without losing important information. Furthermore, because the distance backbone of networks can be computed at a timescale of minutes [15,16], it provides a speed-up of dynamical simulations on larger networks [14]. This is because, as shown here, the distance backbone is the optimal approximation for the dynamics on the overall network.
Despite the performance and explainability of DBS, it has some drawbacks. Currently, measuring the edge importance is done via an exhaustive search, which is computationally expensive. Moreover, like other progressive sparsification methods, DBS requires a practitioner decision on which threshold
to choose for their network analysis. An alternative method with similar performance, but with faster runtime, is the SmD where our multi-objective optimization is achieved for edges with
in this parametric sparsification. For an exact sparsification, our multi-objective optimization has uncovered both lenient and strict sparsification limits as the product and metric backbones [15], respectively, with the latter already proven to best preserve SI dynamics [6]. We found an overall optimal backbone of
, which removes more than half of the connections while preserving more than 80% of the performance metrics. Because
defines the topological space to which edges belong, the optimal
suggests that the DBS parameter could be a characteristic of the fractal topology required for spreading dynamics [37]. This opens up future work to address the extent to which more complex dynamical models, such as Susceptible-Infected-Susceptible and Susceptible-Infected-Recovered epidemics, and synchronization phenomena, require fractal topology.
In the main text, we have focused on DBS under the Dombi path length family (Eq. 11), which generalizes pathfinder networks [26]. However, it is important to note that DBS could be performed using another family of path length measures, as shown in Supplementary Sec. C in S1 Text. Furthermore, for any positive real value of , we are bound to have a higher resolution on edge importance, which our results show to be relevant for inferences on networks. Future work could go beyond distance backbones to its parent theory of distance closures [18] to possibly identify other subgraphs that are capable of preserving a non-shortest-path distance closure.
5. Materials and methods
In this section we describe how the empirical social networks used in our study are built, particularly the strength of interactions within them. Then we explain and compare the network sparsification methods we considered. Finally, we present the definitions of the metrics used to quantify the sparsification performance on node centrality and spreading dynamics.
5.1. Social network construction
We construct empirically measured social contact networks publicly available in previous studies (cited in Table 1), where each node represents a person and people are connected based on the frequency of their physical proximity. The data is from radio frequency identification and Bluetooth devices that accompanied each individual over a certain period of time and registered who was in close physical proximity to whom during short intervals of about 20 seconds. Using to denote the number of intervals between persons
and
during the overall data collection period, and
to denote the total number of close interactions person
had, the strength of interaction between
and
is
This value can be regarded as a similarity measure between two people, or the convention we follow, a proximity measure [15,16,18]. Thus, a graph G has a set of nodes X and weighted edges , which are regarded as the intensity with which two nodes
are connected. From
we measure the isomorphic interaction distance
between nodes using equation 1.
Our approach is easily generalizable to networks beyond the social domain [14]. Table 1 shows the diversity of social-contact-network samples we studied. They cross multiple orders of magnitude in numbers of both nodes and edges, which also suggests the generalizability of our findings. DBS can be used for any weighted graph in which the nodes are connected by a measure of similarity (or dissimilarity) irrespective of the domain. Moreover, because the sample networks are diverse, we expect our recommendations to be domain agnostic. The only common feature among these socially structured networks is the fact that each one has a single minimum spanning tree, as found by their ultrametric backbone size. The extent to which this feature affects our results is left as future work.
5.2. Comparing sparsification methods
Many network sparsification methods define a parameter that quantifies the importance of each edge in a network [7,10,14]. This quantification induces an increasing (or decreasing) sparsification order wherein the least important, and thus removable, edges are those with smaller (or larger) parameter values. We propose that DBS can progressively sparsify networks by removing more edges with increasing values of . We compare DBS performance with other sparsification methods, which we describe below and summarize in Table 2.
The most simplistic way of sparsifying a network is to remove edges that encode weak interactions. This is the case of W, where edges are removed by increasing order of the proximity weight ; lower proximities represent weaker interactions and are removed first.
A more informed approach, the DispFilt [7], assumes a null model for network connectivity to quantify an edge’s statistical significance, . This is used to remove edges in decreasing order because edges with lower
values are statistically more likely to be relevant. More explicitly, DispFilt considers a null model in which the probability density function of a normalized edge weight from a node
taking value
is given by
, where
is the node degree. This leads to
for an undirected network. In other words, an edge is kept if it is significant for at least one of the nodes to which it connects.
Another methodology, which draws inspiration from electrical networks, is wEffRes [10]. It removes edges in increasing order of a measure of weighted resistance between node pairs, , because smaller values are proposed to be less relevant for global information exchange. The effective resistance between any two nodes has information about all the possible paths connecting them because
, where
is the projection of the Moore-Penrose pseudo-inverse of the Laplacian matrix,
, onto the canonical bases
. Usually, computing
is a very difficult task, especially for very large networks, and some approximation algorithm is required [10].
Finally, we also compare DBS with another method that takes into account the metric topology of a weighted graph, SmD [14]. In this method, edges are removed in decreasing order of their semi-metric distortions, , because for spreading dynamics, edges with higher semi-metric distortions have been shown to be less important than those with
[14]. The semi-metric distortion of an edge is given by
where denotes a path.
5.3. Performance metrics
The performance metrics (Fig 5) are compared between the original network and its subgraph with an adjusted relative size, . The number of edges in the subgraph is adjusted with respect to the sizes of the limiting distance backbones subgraphs, the ultrametric
and the drastic Bdrastic. That is, for a network G containing
edges which is sparsified down to the subgraph
, then
Curves for each sparsification method (per legend) are shown for different levels of sparsification, . (Top-Left) Spearman’s rank correlation between the eigenvector centrality of nodes in the sparsified network relative to the original network,
. (Bottom-Left) Time at which half of the nodes become infected in the original network relative to the same time measured in the sparsified network,
. Error bars quantify the standard deviation relative to choosing different seed nodes in the simulations. (Top-Right) Average time in which each node got infected in the original graph considering a specific seed node, horizontal axis, and in networks sparsified by DBS, vertical axis. Each color corresponds to a different sparsification level,
, which is annotated with Spearman’s rank correlation between those times,
. (Bottom-Right) Spearman’s rank correlation between average time of infection,
. Error bars quantify the standard deviation relative to choosing different seed nodes in the simulations.
since, in the studied networks, and
, where X is the set of nodes. Therefore,
for the ultrametric backbone and
for the entire graph.
5.3.1. Node centrality.
Several centrality measures have been proposed to quantify the relevance of each node in a network. In this work, we focus specifically on eigenvector centrality, which is among the most widely used measures of node centrality in complex networks [29]. This centrality measure is very popular because it considers the number of connections a node has as well as their quality, attributing high centrality scores to nodes connected to other high centrality nodes. Hence, each node’s centrality, , corresponds to its entry in the eigenvector associated with the largest eigenvalue,
, of the network proximity adjacency matrix with elements
,
The exact value of is not particularly informative except in comparison to the centrality of the other nodes. Therefore, in order to compare how well a sparsified graph preserves this topological network feature we compute the Spearman’s rank correlation between the node eigenvector centralities in the original network and a sparsified subgraph,
, for different amounts of sparsification
. In the sparsified subgraphs, the proximity weights are the same for the edges kept, and set to zero for the edges removed in order to compute the new centrality. Larger values of
correspond to better preservation of centrality rank.
By introducing an attenuation parameter to Eq. 14 as
we find the Katz centrality [30], a parametric node importance measure which has been shown to be relevant in network analysis [10,31,32]. In Supplementary Fig H in S1 Text we show results for how the performance of DBS changes with .
5.3.2. Spreading dynamics.
In this work, we focus on the SI epidemiological model to simulate spreading dynamics on networks [21]. In this model, a single seed node starts in the Infected state, and the rest of the nodes start in the Susceptible state. Then, at each discrete timestep t, each infected node infects its neighboring nodes with probability , where
corresponds to the proximity value of the edge connecting them and
is a global parameter that regulates the timescale of the spreading. Once all the nodes become infected, the spreading terminates, and the process can be entirely characterized by the time steps at which each node gets infected. We perform these agent-based simulations using 15% of the number of nodes as seed nodes and run 100 simulations for each seed node selected; these conditions are repeated for each sparsification considered. To quantify and compare how the spreading process occurs in the original network and in the sparsified networks, we consider two measures that assess the micro and macro dynamics of each node and the larger population, respectively.
To evaluate how similarly the infection spreads across the networks at the scale of each node, we compare the average order at which the nodes get infected during spreading, . That is, we use the Spearman rank correlation between the time at which nodes get infected in the original graph,
, and in a given sparsified subgraph,
. For a given seed node,
is computed considering the average infection time over 100 simulations and for each amount of sparsification,
. We report the average and standard deviation over the seed nodes.
To estimate the timescale of the infection outbreak, the macro dynamics, we consider the time at which half of the nodes are infected, t0.5, also known as the half-infection time [3]. Although analogous quantities referring to different population proportions could be considered, the half-infection time was shown to be less prone to large fluctuations when compared to the full-infection time (t1) [6]. With this, we aimed at understanding whether the different sparsification methods removed edges that were critical for the global timescale of the infection propagation or not. To appropriately assess this question we considered the ratio between the half-infection time in the original network, , and in a given sparsified subgraph, t0.5, as
the inverse of defined in [14]. This way, we have a bounded and normalized measure,
, apart from small random fluctuations, because the spreading tends to take longer to propagate in sparser networks. For each seed node, we take the median of
over 100 simulations and report the mean and standard deviation over different seed nodes.
5.3.3. Area under the curve summarization.
To quantify which method generally performs better, we compute the area under the curves of ,
, and
for each network up to a sparsification limit
,
With the distribution of this aggregated measure we can evaluate how well a given sparsification method preserves a given performance metric over a reasonable sparsification range in different networks. The AUC will be closer to 1 whenever the given sparsification method allows a greater sparsification range without breaking the network (see 5.2) and does so while maintaining a good performance metric (large values of ,
, or
). Generally, we consider
to be the smallest sparsification that preserves connectivity, however, when this limit is very small (as is the case for DBS), we fix
as a more reasonable sparsification limit size.
5.4. Optimization of performance metrics
To identify the optimal backbone under different scenarios, we find the value of that maximizes the objective function
constrained by for different values of
according to the scenarios in Table 3.
Supporting information
S1 Text. Distance Backbones Optimize Spreading Dynamics and Centrality Ranks in the Sparsification of Complex Networks.
https://doi.org/10.1371/journal.pcsy.0000137.s001
(PDF)
Acknowledgments
We thank the members of the CASCI lab for useful discussions about this project. We also thank Deborah S. Rocha for very thorough line editing, and Joana Loureiro for Author Summary brainstorming.
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