Extending Comparison Methods for 1 Unsigned Networks to Signed Networks

11 We can allow the edges of networks to have both negative and positive weights. For example, signed networks can describe the interactions of microbes. To evaluate the performance of estimators for signed networks, we need quantitative comparison methods for signed networks. Finding such comparison methods is done most easily by extending a comparison method for unsigned networks.

another organism. Previous work (López et al., 2019) has discussed comparing such trophic interaction 43 networks 2 . In such a network signed edge weights are unnecessary because an edge from s 1 to s 2 indicates 44 s 1 promotes the growth of s 2 and s 2 inhibits the growth of s 1 (necessarily because s 2 consumes s 1 ), 45 while swapping the direction of the edge to be from s 2 to s 1 indicates the opposite relationship, that 46 s 2 promotes the growth of s 1 and s 1 inhibits the growth of s 2 (due to s 1 consuming s 2 ). In the signed network framework, the former corresponds to s 1 + → s 2 and s 2 − → s 1 , while the latter corresponds to 48 s 2 + → s 1 and s 1 − → s 2 . Note also that the signed network framework does not require the assumption that 49 the cause of this asymmetric interaction is one organism consuming the other. For example, humans 50 promote the growth of rats because human waste serves as a food source for rats, while rats can inhibit 51 the growth of humans by serving as a vector for human diseases. The signed network framework also other kinds of interactions can be described by the unsigned trophic network framework.

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Previous work has looked at e.g. comparing "local" neighborhoods of nodes in (unweighted) 3 signed 58 networks (Zhu et al., 2017) or diffusion kernels in signed networks (Qi et al., 2008). However, overall the 59 existing literature on signed networks seems underdeveloped. This is surprising because such networks 60 could presumably model an extremely wide range of phenomena, not just ecological interaction networks.

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For example, the "gene interaction networks" defined implicitly in (Mani et al., 2008) could be considered 62 (undirected) networks with mixed-sign edge weights if the edge weights were taken to be the differences 63 2 As pointed out in (López et al., 2019), (unsigned) directed networks can also describe other types of ecological interactions. However, without using signed edge weights, only one type of ecological interaction can be described by a single network, whereas in real ecology all possible types of interaction can occur in the same ecosystem. 3 In the framework of (Tantardini et al., 2019) what (Zhu et al., 2017) addresses is the "unknown node correspondence" comparison problem. In contrast, this paper addresses the "known node correspondence" comparison problem. Other differences are that this paper considers "global" comparisons of entire networks and allows edges to have magnitudes other than 1. about a class of objects with extremely broad potential applicability. If prior literature exists then most 72 likely I was unable to find it due to it using different terminology.

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A "network with mixed-sign edge weights" G with S ∈ N nodes can be identified with 75 (i) a "node set" N G (which we can always assume equals [S] := {1, . . . , S} without loss of generality, 76 cf. the discussion below), and only if there is an edge directed from 78 node s 1 to node s 2 , and 79 (iii) an "edge weight function" For each edge (s 1 , s 2 ) ∈ E G , the value A G (s 1 , s 2 ) is the "weight" of the edge, the value sign(A G (s 1 , s 2 )) = ±1 81 is the "sign" of the edge, and the value |A G (s 1 , s 2 )| is the "magnitude" of the edge. Given an explicit 82 identification between N G and [S] (cf. below), there always exists a unique S × S matrix A G , which is 83 called the "adjacency matrix" of the network G, that corresponds to the edge weight function A G .

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For every pair s 1 , s 2 ∈ N G such that s 1 = s 2 , the definition allows for at most one edge directed from 85 s 1 to s 2 , and for at most one edge directed from s 2 to s 1 . Likewise, for every s ∈ N G , the definition allows 86 for at most one edge ("self-loop") directed from s to s. I.e. "multigraphs" are not considered.

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Because literature already exists on comparing unsigned networks, a practical paradigm for comparing 89 signed networks focuses on how to extend comparison methods for unsigned networks to also apply to 90 signed networks. Herein we will focus only on extending comparison methods for unsigned networks that 91 satisfy a mild "well-behavedness" criterion, termed the "monotonicity principle". Cf. figure 2. At a high level, the idea of the monotonicity principle is that "reasonable" comparison methods 93 for unsigned networks should penalize existence errors no less than they penalize "true misses", i.e.

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situations where an edge is missing in both networks. One can also ask for a "continuous" version of the 95 monotonicity principle whereby the size of the penalty for an existence error must never decrease as the 96 magnitude of the weight of the unmatched edge increases.

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Because the monotonicity principle is a very intuitive property that one would most often tacitly 98 assume to be true by default, we can consider comparison methods for unsigned networks that satisfy 99 the monotonicity principle to be "reasonable". Therefore, to implement our paradigm of extending such 100 comparison methods to also apply to signed networks, we need to be able to check whether any proposed  For all three adversarial attacks (figures 6, 7, and 8) we see that the mixed-sign Spearman correlation 188 is always sensitive to the major changes that occur. At the same time, it is also often able to infer some 189 similarity between the original networks and their attacked counterparts. In many cases this is more 190 reasonable behavior than failing to infer any similarity, or inferring complete "anti-similarity". The  values. In contrast, the raw Spearman correlation is unable to never able to notice any similarity to the

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The results show that attempts to use methods for comparing networks with unsigned edges can lead to 234 useless or even actively misleading results when applied to networks with mixed-sign edge weights. The 235 results also show how methods satisfying the double penalization principle can avoid such results.

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The "whole" of simultaneously considering both the sign and magnitude information in a network with 238 mixed-sign edge weights appears to be "greater than the sum of the parts" of using either the sign 239 information or the magnitude information alone. One way to see this is by noting how the behaviors can investigate whether good responses are also observed for other kinds of adversarial attacks.

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The goal was to consider a large number of networks with varying structure, such that altogether they 291 might be enough to represent all "typical" networks. It could be argued that the chosen distribution of 292 random networks is insufficient to accomplish that goal. See section S6.2.2 for details. For example, it 293 would be interesting to use more edges with magnitudes very different from 1 (cf. e.g. section S5.4.1).

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Future work might investigate whether these results can be replicated using different kinds of networks.

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This work provides no systematic procedure for extending arbitrary (dis)similarity measures to the 296 case of networks with mixed-sign edge weights. The double penalization principle is merely a "design 297 specification". It seems unlikely that such a systematic procedure could exist. Nevertheless, the absence 298 of such a systematic procedure could possibly still be considered to limit the usefulness of this work.

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Herein I identified that trying to compare networks with mixed-sign edge weights using standard network 301 comparison methods can give useless or misleading results. I showed that the extra structure of networks 302 with mixed-sign edge weights means that they need to be compared differently than "standard" networks.

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I identified a unifying principle, the double penalization principle, for identifying methods that are 304 potentially useful for comparing networks with mixed-sign edge weights. I implemented this principle 305 for several methods and showed that one simple method (relative error) already satisfies it "out of the 306 box". This work allows us to directly address the difficulties inherent in making meaningful comparisons 307 between networks with mixed-sign edge weights, rather than ignore those difficulties.