Uncovering hidden dependency in weighted networks via information entropy

Interactions between elements, which are usually represented by networks, have to delineate potentially unequal relationships in terms of their relative importance or direction. The intrinsic unequal relationships of such kind, however, are opaque or hidden in numerous real systems. For instance, when a node in a network with limited interaction capacity spends its capacity to its neighboring nodes, the allocation of the total amount of interactions to them can be vastly diverse. Even if such potentially heterogeneous interactions epitomized by weighted networks are observable, as a result of the aforementioned ego-centric allocation of interactions, the relative importance or dependency between two interacting nodes can only be implicitly accessible. In this work, we precisely pinpoint such relative dependency by proposing the framework to discover hidden dependent relations extracted from weighted networks. For a given weighted network, we provide a systematic criterion to select the most essential interactions for individual nodes based on the concept of information entropy. The criterion is symbolized by assigning the effective number of neighbors or the effective out-degree to each node, and the resultant directed subnetwork decodes the hidden dependent relations by leaving only the most essential directed interactions. We apply our methodology to two time-stamped empirical network data, namely the international trade relations between nations in the world trade web (WTW) and the network of people in the historical record of Korea, Annals of the Joseon Dynasty (AJD). Based on the data analysis, we discover that the properties of mutual dependency encoded in the two systems are vastly different.


I. INTRODUCTION
We observe multitudinous emergent phenomena in our surroundings beyond our expectation: herd behaviors such as bird flocks [1], fish schools [2], and stock market bubbles [3], collective intelligence [4,5], fads [6], and so on. The unexpected and intriguing phenomena stem from collective behaviors of interacting individuals in systems of interest, which is the driving motivation of statistical physics in the first place. In order to elucidate the origins of the phenomena, researchers naturally have paid their attention to interaction structures among the individuals. The interaction among the individuals describes their interrelationships.
One of the most popular and useful ways to understand the relationships is to employ the network representation [7]. Each individual or constituent of a system of interest is called a node or vertex, and pairs of the nodes can be connected via so-called links or edges representing the interactions themselves. The simplest form of network is, of course, composed of binary edges, i.e., each edge exists or not. Despite its simplicity, even such a (literally) simple network representation has taught us a lot about interacting systems and their emergent phenomena, symbolized by a number of crucial concepts such as the degree (the number of neighbors of a node) distribution. Beyond the degree from the act of simply counting the * Corresponding author: lshlj82@gntech.ac.kr neighbors, researchers have discovered and developed more delicate metrics encoding hidden correlations inside networks for better understanding of interacting systems, such as the assortativity (basically the two-point correlation for the degree between interacting nodes) [8,9] and the clustering coefficient (the three-point correlation for the connectivity among node triplets) [10,11], and even higher-order structures [12,13]. Understanding the connectivity structure is important because the structure itself can govern the resultant emergent pattern for a given dynamical rule [14,15].
The aforementioned simple representation as the binary network has led us to a great deal of remarkable discoveries so far, but we have to note that simple networks utilize limited information. What we call an edge or a link in a network corresponds to a rather abstract concept of interaction, which can be vastly diverse. There are two representative ways to move on to overcome the limitation: directed networks by taking the possible asymmetric relation (A → B, but B → A) into account and weighted networks by taking the different quantity of interactions (A ↔ B versus A ⇔ B) into account [7]. Imagine a mobile phone call network describing the level of directionality and intimacy between people. The call data contain information such as the information about identities of callers and receivers, the total number or duration of calls within a given time window, etc. Using the information, we can construct a directed and/or weighted network that details the social relationships much more than its binary counterpart (calling at all or not), where inevitable information loss occurs.
In particular, the directed network representation enables us to find the asymmetric relationship between two nodes, embodied in unidirectional edges. In the above example, we can detect the explicit asymmetry between a node and her friend from the call log, if we obtain the log, of course. However, in the real world, there are many situations where such explicitly revealed directional relations are just out of reach for various reasons. Then, is it possible to uncover the asymmetry or dependency between nodes hidden in networks of interest? We can in fact generalize this process of extracting the hidden asymmetry even further, as the asymmetry is one of the plethora of intrinsic structural correlations in networks. In other words, it is deeply related to the problem of identifying the most essential interactions that govern the whole system that can happen to be asymmetric. In spreading dynamics, for instance, it plays a crucial role as the actual substrate network. Network researchers usually assume the directed network structure to model the potential asymmetry in spreading dynamics, but the directed structure is not always transparent. For instance, in an authoritarian society, opinions of more authoritative people are highly likely to spread to less authoritative people compared to the opposite case, but the authority is usually implicitly assumed. In that case, a part of edges (directed subedges) can participate in the actual spreading dynamics of opinion as modeled in Ref. [16]. This type of hidden pathway in spreading processes on networks is extremely important in epidemic spreading, as demonstrated in the recent coronavirus disease 2019 (COVID- 19) outbreak situation [17,18]. In particular, the contact tracing [19] is reported to be one of the most effective ways to prevent the spreading, so identifying plausible directionality on top of the (undirected) contact network will add much richer information to fight this global pandemic.
In this paper, we propose a systematic framework to extract the most meaningful relationships focused on the asymmetry between connected nodes, i.e., hidden dependency submerged in weighted networks. It consists of the process of extracting the most important neighbors for each node via the concept of the information entropy. This ego-centric viewpoint for each node naturally defines the underlying directionality. We take two real-world weighted networks for our analysis, one from economy and the other from history. The network of international trade between nations and the network of people in an official historical record of Korea show vastly different properties in our framework of extracting the hidden directionality. The effect of concealed asymmetry is much stronger in the former than the latter, which we detail later in regard to their other intrinsic network properties. In particular, through the hidden directionality of the international trade relations, we not only just find a hidden asymmetry, but also provide comprehensive trajectories of the changing reciprocal relation between individual nations over time. We cross-check all of the results and conclusions with the null-model networks generated from randomized weights.
The rest of the paper is organized as follows. We present the procedure of extracting the asymmetric relation and a subnetwork derived from it via the information entropy in Sec. II. To evaluate the dependency of the extracted subnetwork in diverse points of view, we suggest various measures, and show i < l a t e x i t s h a 1 _ b a s e 6 4 = " s I Z X d W R B m a v V s i A L r 6 x x A 2 4 j V 1 I = " > A A A C x H i c j V H L S s N A F D 2 N r 1 p f V Z d u g k V w V Z I q 6 L I g i M s W b C 3 U I k k 6 r U O n S c h M h F L 0 B 9 z q t 4 l / o H / h n X E K a h G d k O T M u f e c m X t v m A o u l e e 9 F p y F x a X l l e J q a W 1 9 Y 3 O r v L 3 T l k m e R a w V J S L J O m E g m e A x a y m u B O u k G Q v G o W B X 4 e h M x 6 / u W C Z 5 E l + q S c p 6 4 2 A Y 8 w G P A k V U k 9 + U K 1 7 V M 8 u d B 7 4 F F d j V S M o v u E Y f C S L k G I M h h i I s E E D S 0 4 U P D y l x P U y J y w h x E 2 e 4 R 4 m 0 O W U x y g i I H d F 3 S L u u Z W P a a 0 9 p 1 B G d I u j N S O n i g D Q J 5 W W E 9 W m u i e f G W b O / e U + N p 7 7 b h P 6 h 9 R o T q 3 B L 7 F + 6 W e Z / d b o W h Q F O T Q 2 c a k o N o 6 u L r E t u u q J v 7 n 6 p S p F D S p z G f Y p n h C O j n P X Z N R p p a t e 9 D U z 8 z W R q V u 8 j m 5 v j X d + S B u z / H O c 8 a N e q / l G 1 1 j y u 1 O t 2 1 E X s Y R + H N M 8 T 1 H G B B l r G + x F P e H b O H e F I J / 9 M d Q p W s 4 t v y 3 n 4 A E g w j 3 A = < / l a t e x i t > j < l a t e x i t s h a 1 _ b a s e 6 4 = " g e N L y 5 s n y 7 A v 0 6 L K 2 u M W 7 k V d G u g = " > A A A C x H i c j V H L S s N A F D 2 N r 1 p f V Z d u g k V w V Z I q 6 L I g i M s W 7 A N q k W Q 6 r W P z I j M R S t E f c K v f J v 6 B / o V 3 x h T U I j o h y Z l z 7 z k z 9 1 4 / C Y R U j v N a s B Y W l 5 Z X i q u l t f W N z a 3 y 9 k 5 b x l n K e I v F Q Z x 2 f U / y Q E S 8 p Y Q K e D d J u R f 6 A e / 4 4 z M d 7 9 z x V I o 4 u l S T h P d D b x S J o W C e I q p 5 e 1 2 u O F X H L H s e u D m o I F + N u P y C K w w Q g y F D C I 4 I i n A A D 5 K e H l w 4 S I j r Y 0 p c S k i Y O M c 9 S q T N K I t T h k f s m L 4 j 2 v V y N q K 9 9 p R G z e i U g N 6 U l D Y O S B N T X k p Y n 2 a b e G a c N f u b 9 9 R 4 6 r t N 6 O / n X i G x C j f E / q W b Z f 5 X p 2 t R G O L U 1 C C o p s Q w u j q W u 2 S m K / r m 9 p e q F D k k x G k 8 o H h K m B n l r M + 2 0 U h T u + 6 t Z + J v J l O z e s / y 3 A z v + p Y 0 Y P f n O O d B u 1 Z 1 j 6 q 1 5 n G l X s 9 H X c Q e 9 n F I 8 z x B H R d o o G W 8 H / G E Z + v c C i x p Z Z + p V i H X 7 O L b s h 4 + A E q Q j 3 E = < / l a t e x i t > l < l a t e x i t s h a 1 _ b a s e 6 4 = " M g S 3 z a y b 2 e 7 k N K 4 U Y B 0 4 U P D y l x P U y J y w h x E 2 e 4 R 4 m 0 O W U x y g i I H d F 3 S L u u Z W P a a 0 9 p 1 B G d I u j N S O n i g D Q J 5 W W E 9 W m u i e f G W b O / e U + N p 7 7 b h P 6 h 9 R o T q 3 B L 7 F + 6 W e Z / d b o W h Q F O T Q 2 c a k o N o 6 u L r E t u u q J v 7 n 6 p S p F D S p z G f Y p n h C O j n P X Z N R p p a t e 9 D U z 8 z W R q V u 8 j m 5 v j X d + S B u z / H O c 8 a N e q / l G 1 1 j y u 1 O t 2 1 E X s Y R + H N M 8 T 1 H G B B l r G + x F P e H b O H e F I J / 9 M d Q p W s 4 t v y 3 n 4 A E 9 Q j 3 M = < / l a t e x i t > m < l a t e x i t s h a 1 _ b a s e 6 4 = " 1 y H S i 8 J G 1 d 5 k k q K I O j e P I F b 1 N v E P 9 C + 8 i V N Q i 2 i G m T k 5 9 5 6 T 3 H u D N B R K e 9 5 r w V l Y X F p e K a 6 W 1 t Y 3 N r f K 2 z s d l W S S 8 T Z L w k R 2 A 1 / x U M S 8 r Y U O e T e V 3 I + C k F 8 F 4 z M T v 7 r j U o k k v t S T l P c j f x S L o W C + J q o V 3 Z Q r X t W z y 5 0 H t R x U k K 9 m U n 7 B N Q Z I w J A h A k c M T T i E D 0 V P D z V 4 S I n r Y 0 q c J C R s n O M e J d J m l M U p w y d 2 T N 8 R 7 X o 5 G 9 P e e C q r Z n R K S K 8 k p Y s D 0 i S U J w m b 0 1 w b z 6 y z Y X / z n l p P c 7 c J / Y P c K y J W 4 5 b Y v 3 S z z P / q T C 0 a Q 5 z a G g T V l F r G V M d y l 8 x 2 x d z c / V K V J o e U O I M H F J e E m V X O + u x a j b K 1 m 9 7 6 N v 5 m M w 1 r 9 i z P z f B u b k k D r v 0 c 5 z z o 1 K u 1 o 2 q 9 d V x p N P J R F 7 G H f R z S P E / Q w A W a a F v v R z z h 2 T l 3 Q k c 5 2 W e q U 8 g 1 u / i 2 n I c P U b C P d A = = < / l a t e x i t > j < l a t e x i t s h a 1 _ b a s e 6 4 = " g e N L y 5 s n y 7 A v 0 6 L K 2 u M W 7 k V d G u g = " > A A A C x H i c j V H L S s N A F D 2 N r 1 p f V Z d u g k V w V Z I q 6 L I g i M s W 7 A N q k W Q 6 r W P z I j M R S t E f c K v f J v 6 B / o V 3 x h T U I j o h y Z l z 7 z k z 9 1 4 / C Y R U j v N a s B Y W l 5 Z X i q u l t f W N z a 3 y 9 k 5 b x l n K e I v F Q Z x 2 f U / y Q E S 8 p Y Q K e D d J u R f 6 A e / 4 4 z M d 7 9 z x V I o 4 u l S T h P d D b x S J o W C e I q p 5 e 1 2 u O F X H L H s e u D m o I F + N u P y C K w w Q g y F D C I 4 I i n A A D 5 K e H l w 4 S I j r Y 0 p c S k i Y O M c 9 S q T N K I t T h k f s m L 4 j 2 v V y N q K 9 9 p R G z e i U g N 6 U l D Y O S B N T X k p Y n 2 a b e G a c N f u b 9 9 R 4 6 r t N 6 O / n X i G x C j f E / q W b Z f 5 X p 2 t R G O L U 1 C C o p s Q w u j q W u 2 S m K / r m 9 p e q F D k k x G k 8 o H h K m B n l r M + 2 0 U h T u + 6 t Z + J v J l O z e s / y 3 A z v + p Y 0 Y P f n O O d B u 1 Z 1 j 6 q 1 5 n G l X s 9 H X c Q e 9 n F I 8 z   the relevant results of two empirical data in Sec. III. We finalize the paper with further discussion is in Sec. IV.

II. EXTRACTION OF DIRECTIONALITY BASED ON THE INFORMATION ENTROPY
We start to present the structural aspect of networks in which we are mainly interested. We exemplify two representatively different cases in Fig. 1. The node i in both panels has the same degree, 4, and the same strength (the sum of the weights on the edges connected to the node), 32, but there is a crucial difference between node i in Fig. 1(a) and that in Fig. 1(b), which is obviously the weight distributions around node i. In other words, it refers to the relative proportion of the same strength, 32, allocated to the edges emanating from node i, which is essentially the cornerstone of our whole investigation. Our main idea is that we can utilize the local or ego-centric distribution of the weights to set the quantitative criterion to pinpoint the most essential neighbors of each node, e.g., j and m in Fig. 1(a) and all of the neighbors j, l, m, and n in Fig. 1(b), which will be shown later to be indeed the case within our framework.
To enlighten the situation a bit more deeply, take a look at the connected node pairs (i, l) and (i, m) in Fig. 1(a). In this example, one can easily guess that the node i has two important neighbors, j and m, and each of nodes l and m has only one important (in fact, the only, so indispensable) neighbor i. There is a crucial difference between the two pairs, however, because the nodes i and m designate each other as an important friend, while in the relation between the nodes i and l, only node l considers node i an important friend and not vice versa. Through such asymmetry from the important friends, we can disclose the one-sided (such as i and l) versus mutual (such as i and m) dependency. We dedicate the subsections in this section to present our step-by-step procedure to quantify this concept of essential neighbors and mutual importance.

A. Normalized weight
Let us consider an undirected and weighted network with N nodes and L edges. For each node, denoted by i, there exists a set of weights on edges connected to its own neighboring nodes {w i j | j ∈ ν(i)} where ν(i) is the set of the neighbors of i, and then the cardinality of ν(i) is the number of the neighbors, or the well-known degree The weighted adjacency matrix W (with its elements w i j for the node pair i and j) is symmetric, i.e., w i j = w ji where w i j > 0 if nodes i and j are connected, and w i j = 0 otherwise. The weight w i j usually represents the quantified level of interaction between i and j, so the fraction of such interaction between i and j in the viewpoint of i corresponds tow where s(i) = j w i j is called the strength of node i in network terminology. We call the weight in Eq. (1) a normalized weight that satisfies jwi j = 1 [20]. Let us regard the strength as the total amount of a node's resources to interact with other. Then the normalized weight implies how much fraction of the interaction level the node partitions to its neighbors for given limited "resources" of interactions. In other words, the normalized weightw i j quantifies the importance of node j from the viewpoint of node i. Notew i j w ji in general even if w i j = w ji due to the different strengths s(i) s( j), which is a conceptual leap presented in this work. Accordingly, the symmetric weighted adjacency matrix W is cast into the asymmetric matrixW with its elementw i j . Therefore, the inequalityw i j >w ji implies that the node j is more important to node i than the other way around.

B. Effective out-degree
Based on the normalized weight defined in the previous subsection, we are ready to set up the scheme to extract the most essential interactions for each node. Note that the normalized weightw i j values for node i in Fig. 1(a) are more heterogeneous than that in Fig. 1(b). Suppose that there are a few dominant neighbors of node i whosew i j values comprise most of the interactions of node i [ Fig. 1(a)]. In that case, we may suggest node i to keep only those dominant neighbors and disregard the rest of less essential neighbors. In contrast, when all of thew i j values are similar [ Fig. 1(b)], we can see that all of the neighbors of node i are almost equally important to node i, so it is natural to keep all of its neighbors. Combining the heterogeneity ofw i j distribution with the fact thatw i j is a probability unit, we employ the information entropy for extracting the most essential neighbors. In Ref. [20], some of the authors of this paper originally introduced such a basic concept of extracting them in weighted networks, and in this paper we rigorously formulate the framework and apply it to real networks to demonstrate its utility.
The normalized weightw i j is basically a probability unit in the set {w i j | j ∈ ν(i)} around node i (e.g., the probability of choosing j out of all of the neighbors of node i if w i j represents the unnormalized proportion of the importance of j to i), so we employ the concept of information entropy to quantify the heterogeneity of the units allocated to each edge attached to the node. In this work, we use the Rényi entropy [21], which is a generalized version of information entropy with a tunable parameter to control the overall sensitivity. The Rényi entropy [21] for node i with the parameter α is given by The thermodynamically relevant (satisfying the additivity) Shannon entropy corresponds to the case of α → 1 [22]. The Rényi entropy S α (i) in Eq. (2) approaches ln k(i) if all of thẽ w i j values are similar, while S α (i) 0 if there exists a single neighbor k dominating the interactions from (note that we emphasize the preposition "from" here-we reveal its importance soon) node i, i.e.,w ik 1. Therefore, we define the effective out-degree (again, note the prefix "out" and "→" in superscript on the symbol in the following formula) of node i by exponentiating S α (i) as which is also known as the Hill number [23] to quantify a diversity of order α or the effective number of species in ecology [24,25]. In Fig. 1, we provide the calculated effective out-degree values below the corresponding cases. As a result of exponentiating, Eq. (3) scales ask → α (i) k(i) for a homogeneous weight distribution andk → α (i) 1 when there exists a single dominant neighbor of i. Using the effective out-degree, we extract the most essential edges by taking the topk → α (i) number of neighbors in the order ofw i j . Most importantly, those essential edges are essential in the viewpoint of i, so the relative importance ofw i j is solely determined from i. This ego-centric approach naturally induces the concept of directionality, which was hidden in the original weighted network and we detail on the core concept of this paper from it in Sec. II C.
The effective out-degree depends not only on the heterogeneity ofw distribution but also on the parameter α for a given distribution ofw. It is known that the Rényi entropy is a nonincreasing function of α regardless of the probability distribution [26], so as a result its exponentiated version, the effective out-degree is also non-increasing as α increases. In particular,k → α (i) = k(i) (it recovers the original degree regardless of thew i j distribution, except for the casew i j = 0 that usually corresponds to the absence of the edge between i and j) for α = 0, whereask → α→∞ (i) = 1/ max j {w i j }, and it satisfies the inequality 1 ≤k → α→∞ (i) ≤ k(i) from 0 <w i j ≤ 1 and |{w i j }| = k(i). This behavior upon the parameter α together with the scaling behavior with respect to the heterogeneity of local weight distribution guarantees that every unisolated node has at least one essential edge in any cases.
In particular, the case of α = 2 is widely used to quantify the heterogeneity [28? -33]. The authors of Ref. [33] actually use 1/k → 2 (in our formalism) to describe the local homogeneity of weights in networks. Yet they focus on extracting the backbone structure by quantifying how peculiar the existence of each weight is compared to the null model, under the assumption of keeping the functional form of original degree distribution. As a result, their approach leads the polar opposite point to ours in the case of uniformly distributed local weights-we keep all of the neighbors because they are equally important, while Ref. [33] does not because they are equally statistically insignificant. This is just a matter of different perspectives, and besides the fact that we use more general values of α in the Rényi entropy, most importantly, we proceed one step further from here to discover the hidden directionality of weighted networks from the next subsection.

C. Construction of a subnetwork with hidden dependency
As we already introduced in Sec. II B, to extract the essential neighbors from the viewpoint of each individual node, we choose only the topk → α (i) neighbors, in the order ofw. We illustrate the process in Fig. 1. Because the calculated effective out-degree is a real number, to practically use it (we need to "cut" the neighbors somewhere) we round offk to the nearest integer K → α ≡ k → α + 0.5 . However, a practical issue can arise if we just apply the K → α without actually looking at the w i j values. Suppose there exist ζ additional neighbors with the same weight as the K → α -th weight in the descending order. Then, it would be unfair if we blindly take only up to the K → α -th weight, because some of the neighbors with exactly the same weights are taken and the others are not from pure luck. In that case, we decide to keep all of such neighbors. Formally, therefore, In the examples in Fig. 1, ζ = 0 so the final integer-valued effective out-degree with α → 1 become κ → α = 2 and 4, respectively. From now on, we refer to this particular integer version of effective out-degree κ → α in Eq. (4). Now it is time to apply this effective out-degree from all of the nodes in a network. In other words, each individual node takes only the neighbors with the top κ → α values of weight from the local weight distribution from the node. Then, we obtain the subnetwork composed of the most essential edges. A crucial phenomenon in this procedure is that for a pair of originally connected nodes i and j, node j may belong to the top κ → α (i) neighbors of node i, but node i may not. In this case, the resultant subnetwork includes the unidirectional edge i → j, but not j → i. This hidden directionality emerges as a result of our local threshold scheme based on information theory, which corresponds to the central theme of this paper.
Mathematically speaking, the subnetwork is represented by the asymmetric binary adjacency matrixÃ α for given α, which gives From the adjacency matrix, the effective in-degree coming from other nodes to node i is also naturally defined as The effective out-degree sets a local threshold assigned to every node to extract a directed backbone structure. In contrast to the global threshold in terms of weight to obtain essential subnetworks for instance, extracting the essential edges with κ → α ensures that not a single node is left out because every node has at least one effective out-degree, as discussed in Sec. II B. Another popular method for backbone extraction is the maximum (or minimum, depending on the definition of the weight) spanning tree (MST) [34] which suffers from the severe restriction of (by definition) tree structure with fixed numbers of edges (one less than the number of nodes). In addition, both the global thresholding and MST cannot extract any directional information that our method naturally yields. Compared to those conventional methods, therefore, our framework of extracting the most essential and potentially directional interactions achieves the goals of finding hidden types of information and not ignoring any nodes' local characteristics at the same time. We use the Shannon entropy as a representative case in the remaining of this paper, so we drop the subscript α → 1 for all the measures from now on, e.g., κ → ≡ κ → α→1 . Note that the statistical method in Ref. [33] can also be used to yield directionality in principle, although Ref. [33] does not actually utilize it, but as we discussed in the last paragraph of Sec. II B, their point of view is different from ours.

D. Mutuality from the normalized weight
One may notice that even before extracting the directed subnetwork, the asymmetry between the normalized weights w i j w ji already insinuates the hidden directionality, which is precisely the topic of this subsection. The simplest measure to quantify the (a)symmetry would be to calculate the Pearson correlation between the normalized weights for opposite directions, which we call mutuality. The mutuality M is thus defined as where µ = N/(2L) is the averaged value ofw i j over all of the connected nodes pairs because each node contributes exactly unity (by the definition of normalized weights) to the total summation composed of 2L connected node pairs. Note that µ = 1/ k , where the mean degree k = 2L/N, which we will use in the forthcoming section. Therefore, Eq. (7) can be recast as which is more practical because one only needs to calculate the pairwise correlation between the normalized weights for opposite directions and the second moment of normalized weights.
The mutuality can be strongly subordinated to the underlying network structure, of course. From the definiton of normalized weights, w i j =w i j s(i) =w ji s( j) in Eq. (1), the inequality between the normalized weightsw i j >w ji is equivalent to s(i) < s( j). The strength tends to increase as the degree increases statistically if we assume the absence of intrinsic nontrivial correlations, so k(i) < k( j) under the same assumption. Thus, one has to note that the mutuality is subject to the "baseline" structural network properties such as the strengthstrength correlation and the degree-degree correlation called the assortativity [8,9], so we already present the mutuality with those baseline measures.

A. Empirical data
We apply the suggested methods to two sets of empirical network data: the world trade web (WTW) [35][36][37][38] and the Annals of the Joseon Dynasty (AJD) [39,40]. Both are timeseries data between 1962 and 2014, and 1392 and 1872, respectively. First, the WTW data is annually recorded and contains the total amount w i→ j of export from a nation i to another nation j, which in turn corresponds to the total amount of import for nation j. We regard each nation as a node and the total amount of export as a weight on the edge from one nation to another. In other words, the WTW is orignially a directed network as w i→ j w j→i in general. As the purpose of the current paper is to reveal the hidden directionality from originally undirected weighted networks, we intentionally construct the undirected (but weighted) version of WTW by assigning an undirected edge with the weight w i j ≡ w i→ j + w j→i as the "trade volume" between two nations. The AJD network data is composed of the relationships between people appearing in a collection of records for historical events in Joseon Dynasty, which is a Korean dynastic kingdom that lasted for approximately five centuries (1392-1897). The network is basically a cooccurrence network, where two people are connected with the weight corresponding to the number of sentences mentioning them together within a ten-year time window. We describe more details in Appendix A. We select the WTW and AJD network data as representative examples that enable us to investigate the temporal evolution of congeneric data. First, let us brief on the most basic constituents of these weighted networks: the distribution of weights themselves and their normalized version. The timestamped distributions of weight and the normalized weight defined in Sec. II A are shown in Fig. 2 (for the readers interested in more basic network measures, we show the degree and strength distributions in Appendix A). Due to the heavy-tailed nature, we show the distributions by means of percentiles as the lower quartile Q1, the median Q2, and the upper quartile Q3, as well as the mean value. Both data show right-skewed distributions, reflected by the large fraction of outliers (the criterion of outliers is defined in the caption of In addition, in the AJD, the distribution ofw looks more heterogeneous than that of w. We believe that a particular characteristic of this data is responsible for it; most w values are concentrated on 1 (i.e., most pairs of people appear only once in the 10-year time windows of AJD: around 80% throughout the entire period) [Fig. 2(c)], but its normalized versionw is split into different valuesw i j = 1/s(i) andw ji = 1/s( j) from various values in {s(i)}. Most of all, the overall or averagedout distributions of w andw investigated at a global (network) level do not offer the hidden directional information we would like to discover, so let us move on to the local distribution in the next subsection, from which we present our core results.
B. Local distribution of the normalized weight and effective out-degree In Sec. II B, we have introduced the concept that the local distribution of the normalized weight around node i denoted by p i (w i j ) yields how many neighbors, i.e., κ → α (i) neighbors defined as Eq. (4), of i are essential among the total k(i) number of neighbors. In other words, the distribution p i (w i j ) determines the effective out-degree κ → α (i), so the overall shape of p i (w i j ) in a network provides an informative clue to predict the κ → α distribution. First, we observe that the two networks show remarkably different distributions of normalized weights. As the degree, which determines the overall scale ofw i j for each i, is inherently heterogeneous [7], we have to rescalew i j first for the overview in an entire network. The left panels of Fig. 3 illustrate the representative distribution p i (z) for each data, where z = (w i j − λ i )/σ i is the rescaled variable with respect to the mean λ i = jwi j /k(i) = 1/k(i) and the standard deviation σ i , by averaging the nonzero values of p i (z) over all of the nodes. The normalized weight distribution of WTW is a typical heavy-tailed distribution observed in many complex interacting systems, while the distribution for AJD is unimodal and well-characterized by its mean λ i and standard deviation σ i . This contrast indicates that the local distribution of weights around individual nodes in WTW is usually much more heterogeneous than that in AJD, as in the situations described in Fig. 1(a) versus 1(b), respectively. Therefore, one can expect that κ → α (i) of most nodes in the WTW will be smaller than their original degree k(i), while most nodes in the AJD will recover their original degrees as the effective out-degrees. The right panels of Fig. 3 confirm such distinct scales of effective out-degrees with respect to the original degrees. The effective out-degrees in the WTW is much more smaller than the original degrees on average [ Fig. 3(b)], while they are almost indistinguishable for the AJD [ Fig. 3(d)]. More specifically, in the WTW even though the number of trading partners of nations usually increases and sometimes fluctuates in time, most nations keep a few important trading partners throughout the period. On the other hand, in the AJD, the average effective out-degree and the average original degree are almost indiscernible throughout the five centuries of Joseon Dynasty. This result verifies the expectation drawn from the local distribution ofw in the left panels of Fig. 3 that there are disproportionately small numbers of essential neighbors compared to the original degree in the WTW, and most neighbors are similarly important (so they are all essential according to our framework) in the AJD.
To investigate the implication of the normalized-weight distribution and the effective out-degrees in the two data further, we generate 100 null-model networks by shuffling the weight w i j in original networks (redistributing the weights uniformly at random to all of the existing edges) and then extract the essential edges according to the procedure described in Sec. II. This shuffling process preserves the degree k(i) for every node, but randomizes everything related to the weight information including the original weight w i j , the strength s(i), and the normalized weightw i j for all of the nodes. We measure the mean effective out-degree of the null-model networks, computed as κ → ran that denotes the the mean effective out-degrees for each null-model network, which are in turn averaged over the 100 null-model networks. As one can clearly see from Fig. 3, the AJD shows no noticeable difference between κ → and κ → ran , while they are systematically differ-ent ( κ → ran is always smaller than κ → ) in the WTW. Again, shuffling the relatively homogeneous normalized-weight distribution of the AJD does not affect the effective out-degrees of the nodes in the AJD much, because the nodes will retrieve most of their original neighbors anyway. In contrast, the fact that the effective out-degrees of randomized version of the WTW are systematically smaller than the real effective out-degrees indicates, as discussed in Fig. 1, that the heterogeneity of link weights around a node is weaker in the real WTW than in the randomized WTW. The shuffling process wipes out any correlation of the link weights around a node and equate the local heterogeneity of link weights with the global-level heterogeneity delineated in Fig. 2(a).

C. Evaluation of dependency
So far, we have investigated the hidden directionality by observing the averaged quantities of the most elementary measures. In this subsection, we take a step further into the systems of interest and suggest a few derivative measures in both global and local levels, to demonstrate the utility of our framework.
As illustrative examples, we show parts of the subnetworks constructed by the procedure in Sec. II C, from the oldest [ Figs. 4

(a) and 4(b)] and latest [Figs. 4(c) and 4(d)]
WTW data (with nontrivial hidden dependency as revealed in previous subsections); in particular, we take ego-centric view of the subnetwork from two characteristic nations, which are

China (CHN) [Figs. 4(a) and 4(c)] and the United States of America (USA) [Figs. 4(b) and 4(d)].
One can see the κ → outgoing edges (pink) and the κ ← incoming edges (gray or green, depending on the reciprocity detailed soon), as defined in Eqs. (5) and (6), respectively. The outgoing and incoming edges here refer to the interaction to trading partner nations that a nation considers essentially important and the interaction from trading partner nations that considers the nation as such, respectively. The intersection of outgoing and incoming edges corresponds to the reciprocal edges (green) that represent the mutually important relations. The effective reciprocal degree κ ↔ (i) = jÃi jÃ ji denotes the number of the reciprocal edges attached to node i, whereÃ i j is an element of the asymmetric binary adjacency matrix in Sec. II C.
Not surprisingly, the enormous growth of the Chinese economy is reflected in the growth in the number of trading partner nations of China (60 → 211) over the decades between 1962 and 2014. In particular, compared to its doubled effective outdegree growth (17 → 34), its effective in-degree has been increased by more than 15 times (12 → 201). As the latter indicates other nations' dependency on China, the dramatic change in κ ← captures each nation's genuine influence to the global economy more accurately than the change in the number of trading partner nations (the original degree). In the case of USA, as expected, it was already one of the most influential nations in 1962 already and it is still the case, and the numbers of its trading partner nations and its effective in-degree are increased supposedly due to the overall economic growth globally. However, at the same time, the effective out-degree of USA has been decreased (33 → 24) during the period. In other words, despite the global economic growth, the international trade of USA has become more heterogeneous among its trading partner nations, which may suggest the global economic inequality. Again, we would like to emphasize that this type of distinct analyses is not possible if we only look at the conventional network measures such as degree, strength, and weight distribution without taking the hidden dependency into account.
To characterize the properties of directed subnetworks from effective out-degrees in more details, we calculate the measures called the relative edge density e and the reciprocity r, defined as respectively. The relative edge density e indicates the fraction of essential neighbors for the nodes in a network on average, or the homogeneity of the local weight distribution. Dividing both the numerator and the denominator in Eq. (9) by N, the effective edge density can be rewritten as e = κ → / k , or the ratio of the mean effective out-degree to the mean degree in the right panels of Fig. 3. The reciprocity r is the fraction of the bidirectional edges among the essential edges. It quantifies the fraction of edges in a weighted network representing the mutually (essentially) dependent relation. We show the temporal changes of e and r for the WTW in the upper panels of Fig. 5 and for the AJD in the lower panels of Fig. 5. For comparison, we also plot the corresponding measures obtained from the weight-shuffled null-model networks introduced in Sec. III B. The relative edge density e in WTW stays at quite a low level roughly between 0.1 and 0.2 [ Fig. 5(a)] with a decreasing trend over time, and the reciprocity r stays around 0.3 [ Fig. 5(b)]. As already mentioned in the previous paragraph, the decreasing values of e is equivalent to the overall increasing trend of k and the relatively flat κ → shown in Fig. 3(b). We can interpret this in such a way that nations take part in the international trade more and more as the world trade expands as time goes by (increasing k over time) but their lion's share of trade is usually dominated by a few number of trading partner nations (the relatively flat κ → over time), yielding the decreasing trend of e.
We clarify the implication of e and r by comparing them with those from the null-model networks. The relative edge density e is larger than that from the null-model networks denoted by [e ran ], but the reciprocity r is smaller than that from the null-model networks denoted by [r ran ], as shown in Fig. 5 (again, [· · · ] indicates the ensemble-averaged quantity). The former is expected because κ → ran < κ → (Fig. 3) and e = κ → / k , so [e ran ] < e. The latter (r < [r ran ]) indicates that the mutually essential reactions happen less likely than a chance. The rank of the weight of a link-trade volume-may be high enough to be counted as effective for one end node but may not for the other, probably caused by the severe disparity in their overall link weights related to the national economic scales. In the randomized version, on the contrary, the links

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Incoming edge (not reciprocal) Outgoing edge Incoming edge (reciprocal) for the weight-shuffled null models. On the right panels, we plot the reciprocity r for the real data and [r ran ] for the weight-shuffled null models.
of every node are assigned weights randomly on equal foots, except for statistical fluctuation, and therefore a link assigned a high weight is likely to be counted as effective for both end nodes.
In contrast, as we have already repeatedly checked, the AJD recovers most of its original interactions as essential ones, i.e., e 1, as shown in Fig. 5(c), which is consistent with the result κ → k in Fig. 3(d). Moreover, the property that most original interactions are recovered in the subnetwork also means that interactions are retrieved in both directions, so the reciprocity r 1 as well, as shown in Fig. 5(d). Simply put, the weights in AJD do not play any significant role due to their near uniformity, which is also confirmed by the observation that the average relative edge density [e ran ] and the average reciprocity [r ran ] of their null-model networks are quite similar to e and r from the real AJD network, as shown in Figs. 5(c) and 5(d). In other words, the weight-shuffling process does not affect the properties of AJD notably, as long as the substrate (binary) network is preserved.
From now on, we apply the concept of the reciprocity learned from the global-level analysis back to the individualnode level, where all of our framework begins in fact. As the "global" version of the reciprocity in Eq. (10) is from the av- eraged measures, we can define its "local" version as which we call the local reciprocity (LR), and it represents how many of essential neighbors of node i also consider node i as their essential neighbor. There is one more thing we introduce as another meaningful measure in the local level, corresponding to the ratio of the effective in-degree to the effective out-degree as which we call the attraction ratio (AR) and describes how attractive node i is to its neighbors, relative to the number of attractive neighbors to node i. Note that there is no global measure corresponding to AR as i κ ← (i) = i κ → (i) trivially, and they always satisfy the inequalities We solely focus on the WTW data here, as not surprisingly for most nodes in the AJD network ρ(i) 1 and τ(i) 1. We show the scatter plot of the local measures defined above from the WTW network in Figs. 6(a) (1962) and 6(b) (2014), where each point represents each nation, and one can easily check the inequality in Eq. (13). The ρ and τ for CHN and USA depicted in Fig. 4 are highlighted by the black empty circles and the arrows. From the scatter plot where the nodes are color-coded with their original degree, one can recognize that nations with many trading partners tend to have large values of ρ and τ, and ρ and τ are positively correlated [ Fig. 6(c)] partly because of the upper bound of ρ for given τ values in Eqs. (13) and (14), we suppose. Naturally, larger values of τ increase the chance for the corresponding trading partner nations that consider the nation as an essential partner to be reciprocal. The correlation is significant throughout the entire period of the data we have examined, as shown in Fig. 6(c).
The locations of nations in this ρ-τ space throughout the time provide an overview of the nations' status in the international trade in terms of their mutual importance to other nations. We take four nations in particular to demonstrate it: CHN, USA, India (IND), and Canada (CAN) and show their temporal changes of LR and AR in Figs. 6(d) and 6(e), respectively. As we have checked in Fig. 4, USA has maintained its theoretically maximum level of LR (ρ = 1: all of USA's essential nations take USA as an essential trading partner all the time) throughout the entire period of the data and its status of the "attractive" (τ > 1) trading partner to other nations with an increasing trend from τ 3.8 to τ 8.0. In the case of CHN, as we have observed in Fig. 4, both AR and LR have been significantly increased for the past few decades, signifying its dramatic economic growth during the period, and one can check that the effect is more substantial for AR (proportional to the number of nations that take China as an important partner).
In particular, the AR seems to augment the distinction between the trading relations in the case of similar values of the LR; For instance, both USA and CAN maintain ρ = 1 (except for the small dip in 1979 for CAN), but the AR for CAN is significantly larger than that for USA throughout the period, i.e., CAN is a much more "attractive" trading partner than USA, relative to the number of nations they respectively take seriously. Taking the different baseline values into account, the temporal trends of AR for the two nations are sim- Another characteristic nation is IND, which shows a decreasing (up to 1980s) and then increasing trend for both LR and AR measures consistent with its recent history of industrial growth [42]. The large rearrangement in the overall international trade in the early 1980s is in fact also observed in the structural change itself, e.g, the connectivity significantly shrank, as shown in Figs. 3(b) and 5(a), which may explain the small dip in CAN as well. The second oil shock [43] occurred during this time may be responsible for this overall reorganization of WTW. In other words, the overall trading capacity was temporarily lowered. Particularly, IND suffered from the reduction of the overall trade volume by the international debit crisis [44] Albeit anecdotally, these examples demonstrate that our method of extracting the hidden dependency provides a unique viewpoint on intricate networked systems. We expect that the effect of the current COVID-19 outbreak on the international trade and global economy can also be analyzed with this type of dependency analysis in the future.

D. Inference to originally directed networks
Let us recall that the original WTW data is composed of directed trade relations: imports and exports for bilateral trading nations, denoted by w i→ j w j→i in general. So far, we have intentionally aggregated the weights as w i j ≡ w i→ j + w j→i regarded as a trade volume between two nations i and j, as a test bed to extract directional information as described in Sec. III A. In this subsection, we finally check if our method has successfully uncovered the genuine directional information by comparing the result to the original data. To recap, there exist the original amount of export from nation i to nation j denoted by w i→ j and the normalized weight from i to j denoted byw i j = w i j /s(i) representing the inferred dependency of i on j. We calculate the Pearson correlation coeffi-cient between w i→ j andw i j when there is the directed edge from i to j in both the original directed network and the directed subnetwork extracted from our method, i.e., when w i→ j 0 and node j belongs the top κ → (i) neighbors of i in terms of weights. From Fig. 7, we can see that the inferred weights in the extracted subnetwork and the real weights in the original directed network are highly correlated, which verifies the validity of our method in estimating the mutual dependency.
The accuracy of this estimation is compared to the case of randomized directed weights w i→ j,ran from the original WTW data, as shown in Fig. 7, where the correlation coefficients represent the comparison between the randomized directed weights w i→ j,ran and the normalized weightsw i j,ran from their own undirected networks by taking the same merging procedure w i j,ran = w i→ j,ran + w j→i,ran . Note that our method regenerates the directional information (the correlation coefficient > 0.4) even in that randomized version to a degree due to the fact that w i j,ran includes the original information w i→ j,ran . However, the correlation is much weaker than the original WTW networks, which implies the randomization process destroys the intrinsic crucial information that our method uses to recover the directionality. Therefore, it indicates both the effectiveness of our method and the amount of hidden information available.

E. Mutuality
As the final analysis, we present the mutuality M in Eq. (7) and compare it with other pairwise correlation measures for structural properties. We have already argued that the mutuality can be subordinated to the underlying network structure in Sec. II D-because the normalized weight is inversely proportional to the degree or the strength when the weights are homogeneous enough or random, the mutuality is expected to be correlated with the degree-degree (D-D) correlation [8,9] or the strength-strength (S-S) correlation. In Fig. 8, we show the temporal changes of those correlation measures for the WTW and the AJD, along with those for their aforementioned nullmodel networks with randomized weights.
First of all, in the case of WTW shown in Fig. 8(a), one can check that the fluctuation of M is much less severe than that of other correlations, in particular, compared to the large fluctuation of the D-D correlation in the late 70s to the early 80s when the substantial reorganization of international trade relations happened as discussed in Sec. III C. In spite of the large structural changes reflected in the large fluctuation in the D-D correlation, the bilateral dependency reflected in M has not been disrupted as severely as the network structure itself, so we speculate the situation as the following: in spite of turmoil in international trades caused by various geopolitical reasons, nations might have tried their best to quickly mitigate the shock and maintain the overall mutual dependence in response.
The implication of mutuality values themselves becomes clear when we compare them with the results from the nullmodel networks. Again, we generate 100 null-model net- works with completely shuffled weights on the original network structure and calculate the mutuality and the S-S correlation (the D-D correlation would be the same because the network structure itself is not altered), and plot their ensembleaveraged values in Fig. 8 in addition to the correlation values from the original networks. The most prominent difference between the original network and its null model is observed in the case of mutuality of the WTW in Fig. 8(a), and in particular, the mutuality of the WTW is much smaller than that of its null model. This again confirms our previous conclusion that the international trade is less mutual, as discussed in Sec. III C and Fig. 5(b). The positive values of M in the case of null models has the same origin as the larger reciprocity discussed in Sec. III B. The average and variance of local link weights are not distinguishable between the two end nodes of a link in the null models. Thus a link with high (low) weight is likely to have similar normalized weights commonly larger (smaller) than the average µ. It does not hold for the real WTW, in which the scales of the link weights of two connected nodes may be quite different, and thus the normalized weight of a link from the viewpoint of one end node may be much different from the other, reducing mutuality. For more discussions and examples, see Appendix B.
The absence of significant effects of weights and the results from it in the AJD is reconfirmed with the mutuality and other correlations as well, as shown in Fig. 8(b). As expected, the mutuality of the AJD is quite similar to that of the corresponding null-model networks, not surprisingly because of their relatively uniform weights, the details of which are already discussed in Sec. III C.

IV. SUMMARY AND DISCUSSION
We have proposed the framework for constructing a directed subnetwork composed of the most essential edges via the concept of information entropy, based on heterogeneity of local distributions of weight around each node. We call the number of such essential neighbors of a node the effective out-degree, which plays the role of a local or ego-centric threshold of extracting the most important neighbors from the node. This naturally appearing but initially hidden directionality from each of individual nodes is the cornerstone of our framework. Although we have focused on the case of the Shannon entropy (α → 1) almost exclusively in our work, by tuning the parameter α one can control the overall sensitivity of the threshold. To demonstrate the utility of our method, we have compared two series of real networks composed of temporal snapshots: the WTW and the AJD, followed by the comparison with their weight-randomized version as the null model. We have analyzed the hidden dependency within the networks by taking both the global-and the local-scale properties and concluded that the WTW has intrinsically less mutual or unequal bilateral dependency between the nations, while people in the AJD are connected with more mutual dependency from their narrowly distributed weights. In addition, we have verified that our method extracts the most essential directed relation by comparing the result with the original directional information (export and import) in the WTW.
We can apply the extracted directed subnetwork to various purposes, depending on the context. In general, the directionality from i to j in our framework indicates the dependency of i on j, so it effectively captures the flow from less influential nodes to more influential nodes, roughly speaking. In social relations, for instance, the directionality may insinuate the hidden authoritative relations among nominally mutual "friendship." Another example is various types of biochemical networks, where seemingly "related" chemical/metabolic reactions or genetic entities could in fact hide their true identity of asymmetric dependency, which would enable us to prioritize a part of networks to engineer the system better, e.g., when we try to find a new drug target. Beyond the inference to the directionality in static networks, we may utilize the fact that the directional information connotes the temporal information as any type of interaction takes time. Therefore, albeit not perfectly, the directionality may help us to deduce the temporal order from temporally accumulated networks as well, which would be of great importance when it comes to reconstruction of causality or the Bayesian formulation [45].
Finally, to take a more concrete example, our analysis of the WTW has demonstrated the potential of our method to applications for economic and other sectors dealing with global problems as well, we believe. We would like to emphasize that pointing out specific nations with characteristic properties in terms of LR and AR is much more meaningful than just providing interesting anecdotal examples, because each of the interrelationships in the WTW actually affects our daily life. The hidden directionality in epidemic spreading processes can be crucial to detect superspreaders or superblockers, which is tightly related to the global economy as now all of us know. We hope to sharpen our tool more to prepare for more practical applications on top of a more solid theoretical background in the future.  because the normalized weights in the opposite direction are always anti-correlated (or completely disassortative [8,9], as they are equivalent here), but r = 1 because the subnetwork retains all of the original edges bidirectionally due to the com-pletely uniform weight values. If we look closely into the situation, we can see that the mutuality solely takes the bilateral relation, while the reciprocity contains the information on the overall weight distributions around each node used to extract the subnetwork.
Therefore, although the mutuality may look intuitive and convenient to use (we need not calculate the entropy measures and others), it is not able to capture a more nuanced concept of mutual dependency: if we take the star network in the previous paragraph again, even if the central node is dominant in the structural aspect (captured by M = −1), all of the peripheral nodes are equally important to the central node as well (captured by r = 1). Of course, there are cases where the former is more relevant depending on the context, so we claim to use both measures to fully characterize a given networked system with weights. We show both measures for our data in Fig. 10, and one can observe that the relationship between two measures is not simple with quite scattered points.