A split-and-transfer flow based entropic centrality

The transfer entropic centrality

Consider the network shown on Fig. 1A and assume that the probability of an indivisible flow going from one vertex to another is uniform at random (including the option to remain at the current vertex). For a flow starting at v₁, there is then a probability $\frac{1}{4}$ to go to v₄, and a probability $\frac{1}{2}$ to continue to v₅, so the probability to go from v₁ to v₅ following the path (v₁, v₄, v₅) is $\frac{1}{8}$. But since it is also possible to reach v₅ from v₁ using v₃ instead, an event of probability $\frac{1}{8}$, we have that the probability p_v1,v5 for an indivisible flow to start at v₁ and stop at v₅ is $p_{v_1,v_5}=\frac{1}{4}$. Similarly, we compute p_v1,v1, p_v1,v2, p_v1,v3 and p_v1,v4, and the transfer entropic centrality C_H(u) of u = v₁ is $C_H\left(v_1\right)=\frac{3}{4}{log}_{2}4+\frac{2}{8}{log}_2\left(8\right)=2.25$ by (1).

For a point of comparison, on the right of the same figure, we change the probability to go out of v₁, such that the edge (v₁, v₂) is chosen with a probability $\frac{1}{2}$, while the probability is $\frac{1}{6}$ for using the edges to the other vertices (including a probability $\frac{1}{6}$ that the flow just stays at v₁ itself). The resulting probabilities are provided on Fig. 1B. There is no complication in computing C_H(v₁) using (1) with non-uniform probabilities. This reduces slightly the centrality of v₁, which is consistent with the interpretation of entropic centrality: the underlying notion of entropy is a measure of uncertainty (Tutzauer, 2007), the uncertainty of the final destination of a flow, knowing that it started at a given vertex. Imagine the most extreme case where the edge (v₁, v₂) is chosen with a probability 1, then even though v₁ has three potential outgoing neighbors, two of them are used with probability 0, so the centrality of v₁ would reduce considerably, as expected, since there is no uncertainty left regarding the destination of a flow at v₁.

The notion of transfer entropic centrality captured by (1) assumes that there is no vertex repetition in the paths taken by the flow. Figure 2 illustrates this hypothesis. Again for the centrality of v₁, a flow leaves v₁, it can go to either v₂, v₃ or v₄. When reaching v₄, the flow cannot go back to v₁, since v₁ is already visited (and going back would not give a path anymore), there the probabilities to stay at v₄ and to go to v₅ from v₄ are modified. On the left, when probabilities are uniform, since now only two outgoing edges of v₄ are available, namely edges (v₄, v₄) and (v₄, v₅), each is assigned a probability of $\frac{1}{2}$. On the right, when probabilities are not uniform, we distribute the probability of going to some visited vertex proportionally to the rest of the available edges. Since $\frac{4}{6}$ is going to v₅ while $\frac{1}{6}$ is staying at v₄, we have 4 and 1 out of 5 respectively leaving and staying, thus obtaining the renormalized probabilities as $\frac{4}{6}+\frac{4}{5}\frac{1}{6}=\frac{4}{5}$ and $\frac{1}{6}+\frac{1}{5}\frac{1}{6}=\frac{1}{5}$.

The examples of Figs. 1 and 2 illustrate diverse cases of indivisible flow. By definition of indivisibility, the choice of an edge at a vertex u corresponds to choosing subsets containing one vertex only in the list of all subsets of neighbors. We can thus set a probability 0 to all subsets which contain more than one vertex. Therefore, the definition of entropic centrality in (1), with or without uniform edge probabilities, are particular cases of the proposed split-and-transfer framework, that we discuss next.

The split-and-transfer entropic centrality

Consider the network of Fig. 3 depicting a seller v₁ whose direct customers are v₂, v₃, v₄. Say we further know that when v₁ distributes a new batch of items, he does so to either customers {v₂, v₃} or {v₃, v₄}, and in fact, the pair {v₃, v₄} is preferred (they receive 2∕3 of the batches, versus 1∕3 for the group {v₂, v₃}). Furthermore, in the first case, v₂ receives a higher volume than v₃ (say 2∕3 of the batch goes to v₂), while for the second case, v₄ takes 3∕4 of the batch shared with v₃. Once v₃, v₄ obtain the items, they typically keep half for themselves, and distribute the other half to v₅.

To compute the centrality of v₁, we consider a divisible flow starting at u = v₁ which can split among different paths instead of following one. To model the choice of splitting among possible neighbors, we first define a probability q(x) over the set ${\mathcal{E}}_u=\left\{\right.\left\{v_1\right\}$, {v₂}, {v₃}, {v₄}, {v₁, v₂}, {v₁, v₃}, {v₁, v₄}, {v₂, v₃}, {v₂, v₄}, {v₃, v₄}, {v₁, v₂, v₃}, {v₁, v₂, v₄}, {v₁, v₃, v₄}, {v₂, v₃, v₄}, $\left\{v_1,v_2,v_3,v_4\right\}\left.\right\}$ such that, for our example, $q\left(\left\{v_2,v_3\right\}\right)=\frac{1}{3}$, $q\left(\left\{v_3,v_4\right\}\right)=\frac{2}{3}$, and q(x) = 0 for other choices of x (in contrast to Oggier, Silivanxay & Datta (2018) where it was chosen to be uniformly at random). This represents the fact that 1∕3 of the time, v₁ sends the goods to the pair {v₂, v₃} (as shown in Fig. 3A), while for the rest of the time, it sends it to the pair {v₃, v₄} (shown in Fig. 3B). We compute the path probabilities for each event, for $q\left(\left\{v_2,v_3\right\}\right)=\frac{1}{3}$ and for $q\left(\left\{v_3,v_4\right\}\right)=\frac{2}{3}$ accordingly.

We have further information: when v₁ deals with {v₂, v₃}, there is a bias of $\frac{2}{3}$ for v₂ compared to $\frac{1}{3}$ for v₃, and the bias is of $\frac{3}{4}$ for v₄ in the other case. The corresponding probabilities are attached to the edges {(v₁, v₂), (v₁, v₃)} and {(v₁, v₃), (v₁, v₄)} respectively (shown in Fig. 3C). Now that the edge probabilities are defined, we can compute the path probabilities. For example, from v₁ to v₅, we sum up the path probabilities for both events, weighted by the respective event probability: $\frac{1}{3}\left(\frac{1}{6}\right)+\frac{2}{3}\left(\frac{1}{8}+\frac{3}{8}\right)$.

We next provide a general formula. We let a flow start at a vertex whose centrality we wish to compute, and at some point of the propagation process, a part f_u of the flow reaches u. Let ${\mathcal{N}}_u$ be the neighborhood of interest given f_u, that is, the set of outgoing neighbors which have not yet been visited by the flow. Every outgoing edge (u, v) of u exactly corresponds to some outgoing neighbor v, so in what follows, we may refer to either one or the other. Let ${\mathcal{E}}_u$ denote the set of possible outgoing edge subsets (where every edge (u, v) is represented by v the neighbor). We attach a possibly distinct probability q(x) to every choice x in ${\mathcal{E}}_u$. Then ${\sum }_{x\in {\mathcal{E}}_u}q\left(x\right)=1$.

Every x in ${\mathcal{E}}_u$ corresponds to a set of edges (u, v) for v a neighbor. We further attach a weight ω_x(u, v) to every edge in x, with the constraint that ∑_(u,v)∈xω_x(u, v) = f_u. For example, we could choose all edges with equal weight, that is ${\omega }_x\left(u,v\right)=\frac{f_u}{i}$ for every (u, v) in x containing i edges, to instantiate the special case where the flow is uniformly split among all edges.

For a given node u, we compute the expected flow from u to a chosen neighbor v. Every such choice of x comes with a probability q(x), and every edge (u, v) in x has a weight ω_x(u, v), which sums up to (2)${\displaystyle f_{uv}=\sum _{x\in {\mathcal{E}}_{u,v}}q\left(x\right){\omega }_x\left(u,v\right),}$ where ${\mathcal{E}}_{u,v}$ contains the sets in ${\mathcal{E}}_u$ themselves containing v.

The property of flow conservation observed in the example holds true in general, which we shall prove next. Indeed, when v varies in ${\mathcal{N}}_u$, the sets ${\mathcal{E}}_{u,v}$ appearing in the summation ${\sum }_{v\in {\mathcal{N}}_u}{\sum }_{x\in {\mathcal{E}}_{u,v}}q\left(x\right){\omega }_x\left(u,v\right)$ may intersect, so for each choice x, one can gather all the ${\mathcal{E}}_{u,v}$ that contains x. For this x, we find a term in the above sum of the form q(x)∑_(u,v)∈xω_x(u, v) = q(x)f_u. Then ${\displaystyle \sum _{v\in {\mathcal{N}}_u}\sum _{x\in {\mathcal{E}}_{u,v}}q\left(x\right){\omega }_x\left(u,v\right)=\sum _{x\in {\mathcal{E}}_u}q\left(x\right)f_u=f_u.}$

This shows that the flow from u to v is conserved over all the neighbors $v\in {\mathcal{N}}_u$ given f_u. Thus, by setting f_u = 1, the quantity ${\displaystyle f_{uv}=\sum _{x\in {\mathcal{E}}_{u,v}}q\left(x\right){\omega }_x\left(u,v\right)}$ becomes a probability, and in fact, putting this probability on the edge (u, v) in the context of the transfer entropic centrality gives the same result as the above computations using the split-and-transfer model, as in fact already illustrated on the figure in Example 1, since the probabilities displayed on the edges of the graph have been computed in this manner. We summarized what we computed in the proposition below.

We thus showed that the split-and-transfer entropic centrality is equivalent to a transfer entropic centrality, assuming the suitable computation of edge probabilities.

The notion of split-and-transfer entropic centrality characterizes the spread of a flow starting at u through the graph. Now two vertices may have the same spread, but one vertex may be dealing with an amount of goods much larger than the other. In order to capture the scale of a flow, we also propose a scaled version of the above entropy.

As a corollary, the computational complexity of this centrality measure is the same as that of Tutzauer (2007), namely that of a depth first search, i.e., O(|V| + |E|) (Migliore, Martorana & Sciortino, 1990). When the graph becomes large and some probability become negligible, a natural heuristic of setting them to 0 is used.

The scaling function F may depend on the nature of the underlying real world phenomenon being modeled by the graph, with F(f_u) = f_u being a simple default possible choice (other standard choices are $F\left(f_u\right)=\sqrt{f_u}$ or F(f_u) = log(f_u)). We use the default choice in the example below.

Shareholding in Tehran Stock Market

We next consider 131 companies from the Tehran Stock Market, as listed in Appendix A of Dastkhan & Gharneh (2016).¹ They form the vertices of a cross-shareholding network of companies which have shares of other companies. There is an edge between i and j if company i belongs to the investment portfolio of company j, i.e., j owns some share of i. Therefore the in-degree of node j is the number of companies that belong to the investment portfolio of company j. Conversely, the out-degree of node j is the number of companies that are shareholders of j. Edges are weighted, edge (i, j) has for weight the percentage of shares that company j has from company i. We will consider this graph, shown in Fig. 4, and the graph with reverse edge directionality.

Nodes highlighted in green in Fig. 4 have one edge with weight more than 0.5, meaning that more than 50% of their shares are owned by another company, otherwise they are in grey. Nodes highlighted in vermillion, superseding the other coloring, have the highest in-degrees, which means that they own shares of many other companies. They are nodes 121 (Adashare), 85 (NIKX1), 123 (Oilcopen), 42 (SA3A1), 119 (tamin org), and 118 (government). Nodes highlighted in sky blue have the highest out-degrees, which means that their shares are owned by many other companies (but possibly in small amounts). They are nodes 110 (BMEX1), 2 (CH12), 21 (FO041), 41 (GD021), 3 (GO02) with degree 7 and 69 (PFAX1), 1 (MADN), 20 (MS022) and 62 (PK061) with degree 8.

We next assign probabilities to edges: we use the edge weight, and fix the edge probability to be inversely proportional to its weight. Self-loops have a natural interpretation. Since the outgoing edges of node j indicate the companies that are shareholders of j, the self-loop refers to j still owning some of its own shares, and the amount is 100% minus what the other companies own (share ownerships with negligible amounts were not taken into account in the data set, so self-loops absorb these portions).

Table 1 lists the seven nodes with the highest entropic centrality. The interpretation of entropic centrality here is that we are looking at the nodes whose shares are “most diversely owned” in terms of their shares being owned by different companies, whose shares are in turn themselves owned by others. The economic fortunes of the company whose centrality is looked at thus also affects those of the other companies, and the entropic centrality thus indicates the impact a particular company’s economic performance would have on the rest. We immediately see that this centrality measure is different from out-degree centrality. We can however look at how they relate, by considering the role of out-degrees not only on the nodes but also on their neighbors. We observe that only node 3 has one of the highest out-degrees, however, nodes 23 and 116 still have high out-degrees, but also are connected to neighbors which have high out-degrees, in fact, node 23 which has the highest entropic centrality has three high out-degree neighbors. For nodes 109 and 59, they still have a relatively high out-degree. For 5, it has a fairly low out-degree, but out of the 3 neighbors, two have high degrees themselves. The case of node 65 is particularly interesting, since it has only two neighbors, namely 55 and 57. Neither of 55 nor 57 has a high centrality individually, but they together provide node 65 a conduit to a larger subgraph than the individual transit nodes themselves do, illustrating the secondary effects of flow propagation.

The cross-shareholding network of Tehran Stock Market was analyzed in Dastkhan & Gharneh (2016), where a closeness centrality ranking is shown to be almost identical to the degree based centrality one. The entropic centrality ranking in contrast manages to capture a different dynamics, by involving the spread of influence via flow propagation, together with a quantitative edge differentiation.

The above approaches ignore any other information such as the market size share of the organizations. Arguably, between two organizations with identical position in the graph, the one with larger market size may be deemed to have larger influence on the other nodes. This is modeled by the scaled entropic centrality C_H(u)f_u, where f_u is the market size in percentage. We notice that this indeed creates a distinct relative ranking (indicated by subscript in Table 1, for example, ARFZ1 is ranked 3rd as per weighted entropic centrality). Particularly, among the top seven companies as per C_H(u), we see that only PRDS1 continues to be in the same (fourth) rank. KNRX has the largest change in ranking, up from sixth to second. While the scaled entropic centrality ranking of the top two nodes are congruent with the ranking based solely on the scale factor (market size), we see that ARFZ1, which would be ranked 5th by market size, and first by solely network effect, is ranked 3rd when both factors are taken into account.

We compare the entropic centrality C_H,p with respect to the alpha, Katz and PageRank centralities (using the reverse edge direction). Note that the unweighted graph has for largest eigenvalue λ₁ ≃ 2.99715780 (so $\frac{1}{{\lambda }_1}\simeq 0.3336494$). The ranking results for the most central entities from the Tehran stock exchange, and the overall Kendall tau rank correlation coefficients Kendall (1938) are reported in Tables 1 and 2 respectively. The Kendall tau coefficient indicates the rank correlation among a pair or ranked lists (see Schoch, Valente & Brandes, 2017 for a discussion on why Kendall tau is preferred to Pearson). The entity 23 is outstandingly central with respect to all metrics but weighted betweenness. The alpha and Katz centralities yield very similar results, but they rank the entity 3 as third instead of second. They rank second the entity 20. A likely explanation could be that that 20 actually has a higher out-degree (and thus a higher in-degree in the reversed edge network) than entity 3. The top 7 most central entities have mostly a 0 betweenness (ranked 39), and are mostly ranked pretty low with respect to both versions of PageRank, weighted and unweighted. The most central entity for the weighted betweenness is 85, which is one of the most central with respect to in-degree (it has an in-degree of 18). Then 111 and 76 are second respectively for the unweighted and weighted PageRank. Entity 111 has out-neighbors 88,81,127,94,112 which become in-neighbors in the reversed edge graph, neither 111 itself nor its neighbors stand out by their degrees, however 76 has for in-neighbors in the reversed edge graph 72,73,130,85,118,76, and both 118 and 85 are very central with respect to in-degree, making it easier to explain why it is ranked high. Note that the assortativity coefficient of this graph is ≈ − 0.01521584, so this is a non-assortative graph, where high degree nodes do not particularly connect to neither high nor low degree nodes.

The Kendall rank correlation confirms that the entropic centrality differs from the other metrics not only to decide the most central vertices but also overall. The Kendall coefficient for unweighted betweenness has not been reported since only 38 vertices have a non-zero betweenness. This shows that the graph considered is far from being strongly connected. Overall, comparison points illustrate that the entropic centrality C_H,p provides a new perspective not captured by the other algorithms.

We next consider the same graph but where edge directionality is reversed. A node becomes central if it owns diverse companies, which themselves may in turn own various companies. Since owning shares could be used to influence an organization’s management, the entropic centrality based on reverse edges is thus a proxy indicator of how much control a specific entity has over the other entities in the market. Organizations with very high entropic centralities using either sense of edge direction could then be seen as probable candidates causing structural risks—be it by being ‘too big to fail’, or having too much control over significant portions of the market for it to be fairly competitive.

The nodes with highest entropic centrality are shown in Table 1. The most important one is the government: we expect it to be one of the most important players in Iran when it comes to owning shares in other companies (and yet not to appear in the list when the original edge direction is considered). We see a higher correlation between entropic centrality and in-degree than there was between entropic centrality and out-degree in Table 1. Among the seven most central nodes, five of them are having one of the highest in-degrees, the two most central nodes have themselves one high degree neighbor. Then node 88 has a fairly low in-degree, but it is connected to node 85.

In summary, the case study of the Tehran stock exchange network exhibits three important intertwined aspects of our model. Firstly, it is flexible. It seamlessly captures the effect of relationships, considered either in a binary fashion (just the structure), or quantified with relative strengths (the skew in strength of the relationships), while it can also accommodate information which is intrinsic to the node but somewhat disentangled from the network (used as a scale factor). Second, reversing the edge direction gives a dual perspective. Finally, we see that we obtain different results and corresponding insights, based on which variations of the model is applied for a specific study. Naturally, figuring it out the best variation is done on a case-by-case basis.

A bitcoin subgraph

Our final case study is a subgraph of the Bitcoin wallet address network derived from Bitcoin transaction logs (see Fig. 5). Bitcoin is a cryptocurrency (Nakamoto, 2008), and transactions (buy and sell) among users of this currency are stored and publicly available in a distributed ledger called blockchain. User identities are unknown, but each user has one or many wallet addresses, that are identifiers in every transaction. Then one transaction record amalgamates the wallet addresses of possibly several payers and payment receivers, together with the transaction amounts.

To be more precise, consider two Bitcoin transactions T₁ and T₂. The transaction T₁ has n inputs, from wallet addresses A₁, …, A_n, of amounts i₁₁, …, i_1n respectively. The outputs, of amounts o₁₁, …, o_1m go to wallet addresses C₁, …, C_m respectively. The sum of inputs equals the sum of outputs and any transaction fees (say τ₁), i.e., $\left|T_1\right|={\sum }_{k=1}^n{i}_{1k}={\tau }_1+{\sum }_{l=1}^m{o}_{1l}$. For the sake of simplicity, we will ignore the transaction fees (i.e., consider τ₁ = 0). A similar setting holds for transaction T₂, where the same wallet address A₁ appears again as part of the inputs, together with some wallet addresses B₂, …, B_n′ which may or not intersect with A₂, …, A_n. By design, Bitcoin transactions do not retain an association as to which specific inputs within a transaction are used to create specific outputs.

Suppose one would like to create a derived address network from some extract of the Bitcoin logs of transactions, that is a graph whose nodes are Bitcoin wallet addresses, and edges are directed and weighted. There should be an edge from address u to address v if there is at least one transaction where some amount of Bitcoin is going from u to v. However as explained above, it is not always possible to disambiguate the input–output pairs. If the input amounts are particularly mutually distinct, and so are the output amounts, and there are input–output amounts that match closely, one might be able to make reasonable guess about matching a specific input to a specific output. In general, in absence of such particular information, one heuristic is to model the input–output association probabilistically. A common heuristic (Kondor et al., 2014) is to consider that based on transaction T₁ there is an edge from A₁ to each of C₁, …, C_m. The same holds for transaction T₂. Thus in the derived address network, there will be an edge from A₁ to each of the C₁, …, C_m, D₁, …, D_m′. If some outputs X, …, Z are in common to both transaction outputs, there is a single edge between A₁ and each of the addresses X, …, Z.

The derived address network gives us the graph to be analyzed, whose vertices are wallet addresses and edges are built as above. Originally, a given wallet address is sending Bitcoins to possibly different output wallet addresses within one transaction, and the same wallet address may be involved in different transactions, with possible reoccurences of the same output addresses (this is the case of A₁ which is an input to both transactions T₁ and T₂ and X, …, Z appear as output transactions in both). In the split-and-transfer flow model, we can incorporate this information into the derived address network by assigning the probabilities $q\left(\left\{C_1,\dots ,C_m\right\}\right)=\frac{i_{11}}{i_{11}+i_{21}}$ and $q\left(\left\{D_1,\dots ,D_{m^{\prime }}\right\}\right)=\frac{i_{21}}{i_{21}+i_{22}}$ with which the respective subsets {C₁, …, C_m} and {D₁, …, D_m′} of the set {C₁, …, C_m, D₁, …, D_m′} of neighbors of A₁ are used. Other choices for q(x) are possible, the rationale for this specific choice is to use a probability that is proportional to the amount of Bitcoin injected by A₁ in each of the transactions.

Edge weights in the derived address network are computed as follows. Let $\left|T_1\right|={\sum }_{k=1}^m{o}_{1k}$ and $\left|T_2\right|={\sum }_{k=1}^{m^{\prime }}{o}_{2k}$ denote the total amounts involved in each of the transactions. For an edge between A₁ and C_l, which happens in T₁, it is ${\omega }_{C_1,\dots ,C_m}\left(A_1,C_l\right)=\frac{o_{1l}}{\left|T_1\right|}$, while for an edge between A₁ and D_l, which happens in T₂, it is ${\omega }_{D_1,\dots ,D_{m^{\prime }}}\left(A_1,D_l\right)=\frac{o_{2l}}{\left|T_2\right|}$. We thus have ${\displaystyle \sum _{l=1}^m{\omega }_{C_1,\dots ,C_m}\left(A_1,C_l\right)=\sum _{l=1}^m\frac{o_{1l}}{\left|T_1\right|}=1,\sum _{l=1}^{m^{\prime }}{\omega }_{D_1,\dots ,D_{m^{\prime }}}\left(A_1,D_l\right)=\sum _{l=1}^{m^{\prime }}\frac{o_{2l}}{\left|T_2\right|}=1.}$ If some node pairs, and thus edges, repeat across transactions (for example, A₁ to X, …, Z in our example), these edge weights should cumulate in the overall derived address network. This is captured by the formula (2) which is here instantiated as ${\displaystyle f_{A_1,X}=q\left(\left\{C_1,\dots ,C_m\right\}\right){\omega }_{C_1,\dots ,C_m}\left(A_1,X\right)+q\left(\left\{D_1,\dots ,D_{m^{\prime }}\right\}\right){\omega }_{D_1,\dots ,D_{m^{\prime }}}\left(A_1,X\right)}$ ${\displaystyle =\frac{i_{11}}{i_{11}+i_{21}}\frac{o_{1x}}{\left|T_1\right|}+\frac{i_{21}}{i_{11}+i_{21}}\frac{o_{2x}}{\left|T_2\right|}}$ where in transaction T₁ the output to address X is o_1x, while it is o_2x in transaction T₂.

In a departure from previous works that derive the address network in a manner explained above Kondor et al. (2014), our graph model is able to retain the information that subsets of edges co-occur, or not, as displayed above. For that reason, the Bitcoin address network is a natural candidate (and in fact, part of the inspiration) for the general flow model with arbitrary splits and transfers as considered in this paper, where individual flows may go through a subset of outgoing edges.

For our experiments, we choose a Bitcoin subgraph appearing in the investigation of wallet addresses involved in extorting victims of Ashley-Madison data breach (see Oggier, Phetsouvanh & Datta, 2018a for accessing the data). It is obtained by extracting a subgraph of radius 4 (if the graph were undirected) around the wallet address 1G52wBtL51GwkUdyJNYvMpiXtqaGkTLrMv. While we would like to emphasize that we use here this Bitcoin graph to explore the entropic centrality model, it may still be worth mentioning that one identified suspect node from another of our study (Phetsouvanh, Oggier & Datta, 2018), namely node 15hWpb3m5VXdn9KVsS4rSMnrQQJLhXVyN4, has high enough entropic centrality to be listed (see Table 3 below) as a top entropic centrality node. Thus, the entropic centrality analysis can be used as a tool to identify nodes of interest, and to create a shortlist of nodes to be investigated further in detail, in the context of Bitcoin forensics.

Tables 3 and 4 compare the entropic centrality C_H,p with other centralities. With respect to scaled entropic centrality, there is a large variation in the weightages associated with the edges, which has a significant impact on the relative rankings between scaled/unscaled entropic centralities. With respect to weighted betweenness, only three addresses are relevant, they are, with their respective in- and out-degree, 3Eab4nDg6WJ5WR1uvWQirtMzWaA34RQk9s (ranked 1, in-degree: 196, out-degree: 568), 38PjB1ghFrD9UQs7HV5n15Wt1i3mZP8Wke (ranked 2, in-degree: 218, out-degree: 382), and 3CD1QW6fjgTwKq3Pj97nty28WZAVkziNom (in-degree: 14, out-degree: 2807). The other addresses are ranked 69 (corresponding to a betweenness of 0). The graph has for largest eigenvalue λ₁ ≃ 7.1644140 and $\frac{1}{{\lambda }_1}\simeq 0.139578$. As with the previous cases, alpha and Katz centralities are very close to each other, they also agree more closely with C_H,p on the most central addresses, but Table 4 shows that this is not the case in general. The trends shown by the Kendall rank correlation coefficient is similar to previous cases: there are more dissimilarities between PageRank and entropic centralities than between alpha/Katz and entropic centralities, but overall, entropic centralities give a different view point, as would be expected by extrapolating Borgatti’s view point. The assortativity coefficient of the illustrated Bitcoin subgraph is ≈ − 0.11914239, suggesting a slight disassortativity. This is easily explained as an artefact of the way the subgraph was extracted (a small radius around a node), yielding a couple of hubs with nodes connected only to them (leaves). In this example, these leaves are having an entropic centrality influenced by having these hubs as their first neighbors.

As a last scenario, we consider the small network of Maine airports, with their connecting flights, for a total of 55 airports (see Oggier, Phetsouvanh & Datta, 2018b for accessing the data.). We created the network based on flights involving passenger for the period of January 2018 as per the data obtained from the United States Department of Transportation Bureau of Transportation Statistics website (https://www.transtats.bts.gov/DL_SelectFields.asp?Table_ID=292).

In Table 5, we synopsize the Kendall’s tau coefficient τ_κ. The lower the value of this coefficient, the closer (similar) two ranked lists are. We see that C_H,unif produces results which are very similar to alpha and Katz centralities, but C_H,p yields a reasonably distinct result instead. Furthermore, results from both PageRank applied to both weighted and unweighted graphs are most distinct both with respect to entropic centralities, as well as the other existing centralities explored in our experiments. The assortative coefficient is ≈ − 0.71478751, this is thus a disassortative network. Indeed it contains two airports that serve as hubs, and small airports connected to it (or important airports whose edges have been cut when extracting the specific subgraph). In terms of entropic centrality, this corresponds to having small airports inheriting the influence of being connected to hubs.